MEI STRUCTURED MATHEMATICS
EXAMINATION FORMULAE AND TABLES
1
Arithmetic series
General (kth) term,
last (nth) term, l =
Sum to n terms,
Geometric series
General (kth) term,
Sum to n terms,
Sum to infinity
Infinite series
f(x)
uk = a + (k – 1)d
un = a + (n – l)d
Sn = 1–2 n(a + l) = 1–2 n[2a + (n – 1)d]
uk = a r k–1
a(1 – r n)
a(r n – 1)
Sn = –––––––– = ––––––––
1–r
r–1
x2
xr
= f(0) + xf'(0) + –– f"(0) + ... + –– f (r)(0) + ...
2!
r!
f(x)
(x – a)2
(x – a)rf(r)(a)
= f(a) + (x – a)f'(a) + –––––– f"(a) + ... + –––––––––– + ...
r!
2!
f(a + x)
x2
xr
= f(a) + xf'(a) + –– f"(a) + ... + –– f(r)(a) + ...
2!
r!
x2
xr
ex = exp(x) = 1 + x + –– + ... + –– + ... , all x
2!
r!
a ,–1<r<1
S∞ = –––––
1–r
x2
x3
xr
= x – –– + –– – ... + (–1)r+1 –– + ... , – 1 < x 1
2
3
r
sin x
x3
x5
x 2r+1
= x – –– + –– – ... + (–1)r –––––––– + ... , all x
3!
5!
(2r + 1)!
cos x
x2
x4
x 2r
= 1 – –– + –– – ... + (–1)r –––– + ... , all x
2!
4!
(2r)!
arctan x
x3
x5
x 2r+1
= x – –– + –– – ... + (–1)r –––––– + ... , – 1 x 1
3
5
2r + 1
General case
sinh x
n(n – 1) 2
n(n – 1) ... (n – r + 1) r
(1 + x)n = 1 + nx + –––––––
x + ... + –––––––––––––––––
x + ... , |x| < 1,
2!
1.2 ... r
n∈
x3
x5
x 2r+1
= x + –– + –– + ... + –––––––– + ... , all x
3!
5!
(2r + 1)!
cosh x
x2
x4
x 2r
= 1 + –– + –– + ... + –––– + ... , all x
2!
4!
(2r)!
artanh x
x3
x5
x 2r+1
= x + –– + –– + ... + –––––––– + ... , – 1 < x < 1
3
5
(2r + 1)
Binomial expansions
When n is a positive integer
n
n
n
(a + b)n = an + 1 an –1 b + 2 an–2 b2 + ... + r an–r br + ... bn , n ∈ where
n
n
n
n+1
n!
n
+ r+1 = r+1
r = Cr = ––––––––
r
r!(n – r)!
()
()
()
()
() ( ) (
)
2
Logarithms and exponentials
exln a = ax
logbx
loga x = –––––
logba
Numerical solution of equations
f(xn)
Newton-Raphson iterative formula for solving f(x) = 0, xn+1 = xn – ––––
f'(xn)
Complex Numbers
{r(cos θ + j sin θ)}n = r n(cos nθ + j sin nθ)
ejθ = cos θ + j sin θ
2πk j) for k = 0, 1, 2, ..., n – 1
The roots of zn = 1 are given by z = exp( ––––
n
Finite series
n
n
1
1
∑ r2 = – n(n + 1)(2n + 1)
∑ r3 = – n2(n + 1)2
4
6
r=1
r=1
ALGEBRA
ln(1 + x)
Hyperbolic functions
cosh2x – sinh2x = 1, sinh2x = 2sinhx coshx, cosh2x = cosh2x + sinh2x
arcosh x = ln(x +
x 2 + 1 ),
1
+
x
1
artanh x = –2 ln –––––
1 – x , |x| < 1
arsinh x = ln(x +
(
x 2 – 1 ), x 1
)
Matrices
Anticlockwise rotation through angle θ, centre O:
Reflection in the line y = x tan θ :
θ
( cos
sin θ
2θ
( cos
sin 2θ
–sin θ
cos θ
)
sin 2θ
–cos 2θ
)
Cosine rule
A
b2 + c2 – a2 (etc.)
cos A = ––––––––––
2bc
c
a2 = b2 + c2 –2bc cos A (etc.)
B
Trigonometry
Perpendicular distance of a point from a line and a plane
b
a
Line: (x1,y1) from ax + by + c = 0 :
a2 + b2
n1α + n2β + n3γ + d
Plane: (α,β,γ) from n1x + n2y + n3z + d = 0 : ––––––––––––––––––
√(n12 + n22 + n32)
C
sin (θ ± φ) = sin θ cos φ ± cos θ sin φ
cos (θ ± φ) = cos θ cos φ sin θ sin φ
Vector product
i a1 b1
a2b3 – a3b2
^
j
a
b
a × b = |a| |b| sinθ n =
2 2 = a3b1 – a1b3
k a3 b3
a1b2 – a2b1
tan θ ± tan φ
tan (θ ± φ) = –––––––––––– , [(θ ± φ) ≠ (k + W)π]
1 tan θ tan φ
|
a × (b × c) = (c . a) b – (a . b) c
sin θ – sin φ = 2 cos 1–2 (θ + φ) sin 1–2 (θ – φ)
Conics
1–
2 (θ – φ)
1
– (θ – φ)
2
Ellipse
Parabola
Hyperbola
Rectangular
hyperbola
Standard
form
y2
x2 + ––
––
=1
2
b2
a
y2 = 4ax
y2
x2 – ––
––
=1
2
b2
a
x y = c2
Parametric form
(acosθ, bsinθ)
(at2, 2at)
(asecθ, btanθ)
(ct, –c–)
t
3
Vectors and 3-D coordinate geometry
(The position vectors of points A, B, C are a, b, c.)
The position vector of the point dividing AB in the ratio λ:µ
µa + λb
is –––––––
(λ + µ)
Line:
)
a1 b1 c1
a. (b × c) = a2 b2 c2 = b. (c × a) = c. (a × b)
a3 b3 c3
(1 + t )
sin θ + sin φ = 2 sin 1–2 (θ + φ) cos 1–2 (θ – φ)
cos θ + cos φ = 2 cos 1–2 (θ + φ) cos
cos θ – cos φ = –2 sin 1–2 (θ + φ) sin
| |(
|
Cartesian equation of line through A in direction u is
x – a1
y – a2
z – a3
=t
= ––––––
= ––––––
––––––
u1
u2
u3
( )
e<1
= a2 (1 – e2)
e=1
Foci
(± ae, 0)
(a, 0)
(± ae, 0)
(±c√2, ±c√2)
Directrices
x = ± –a
e
x = –a
x = ± –ae
x + y = ±c√2
Asymptotes
none
none
y
x
a– = ± –b
x = 0, y = 0
Eccentricity
b2
a.u
The resolved part of a in the direction u is –––––
|u|
Any of these conics can be expressed in polar
coordinates (with the focus as the origin) as:
where l is the length of the semi-latus rectum.
Plane: Cartesian equation of plane through A with normal n is
n1 x + n2y + n3z + d = 0 where d = –a . n
Mensuration
The plane through non-collinear points A, B and C has vector equation
r = a + s(b – a) + t(c – a) = (1 – s – t) a + sb + tc
The plane through A parallel to u and v has equation
r = a + su + tv
Cone :
b2
Sphere : Surface area = 4π r2
Curved surface area = π r × slant height
e>1
= a2 (e2 – 1)
e = √2
l
– = 1 + e cos θ
r
TRIGONOMETRY, VECTORS AND GEOMETRY
(1 – t2)
2t , cos θ = ––––––
For t = tan 1–2 θ : sin θ = ––––––
2
2
(1 + t )
ax1 + by1 + c
Differentiation f(x)
tan kx
sec x
cot x
cosec x
arcsin x
f'(x)
ksec2 kx
sec x tan x
–cosec2 x
–cosec x cot x
1
–––––––
√(1 – x2)
–1
–––––––
√(1 – x2)
1
–––––
1 + x2
cosh x
sinh x
sech2 x
1
–––––––
√(1 + x2)
arccos x
arctan x
sinh x
cosh x
tanh x
arsinh x
artanh x
4
du
dv
v ––– – u –––
dx
dx
u dy
Quotient rule y = – , ––– =
2
v
v dx
Trapezium rule
b–a
∫a ydx ≈ 1–2 h{(y0 + yn) + 2(y1 + y2 + ... + yn–1)}, where h = –––––
n
b
Integration by parts
Area of a sector
Arc length
dv
du
dx = uv – ∫ v ––– dx
∫ u –––
dx
dx
A = 1–2 ∫ r dθ (polar coordinates)
A = 1–2 ∫ (xẏ – yẋ) dt (parametric form)
s = ∫ √ (ẋ + ẏ ) dt (parametric form)
dy
s = ∫ √ (1 + [ –––] ) dx (cartesian coordinates)
dx
dr
s = ∫ √ (r + [ –––] ) dθ (polar coordinates)
dθ
2
2
2
2
2
2
sec x
1
––––––
2
x – a2
1
–––––––
√(a2 – x2)
1
––––––
a2 + x2
1
––––––
a2 – x2
sinh x
cosh x
tanh x
1
–––––––
√(a2 + x2)
1
–––––––
√(x2 – a2)
Surface area of revolution
∫f(x) dx (+ a constant)
(l/k) tan kx
ln |sec x|
ln |sin x|
x
–ln |cosec x + cot x| = ln |tan –2 |
x π
ln |sec x + tan x| = ln tan – + –
2 4
1
x–a
––
ln –––
x +––
a
2a
| (
| |
arcsin ( –ax ) , |x| < a
1– arctan –x
( a)
a
1
1
x
a+x
––
ln ––––– = – artanh ( –a ) , |x| < a
2a | a – x | a
cosh x
sinh x
ln cosh x
arsinh –ax or ln (x + x 2 + a 2 ),
()
arcosh ( –ax ) or ln (x + x
S = 2π∫y ds = 2π∫y√(ẋ
S = 2π∫x ds = 2π∫x√(ẋ
2
– a 2 ), x > a , a > 0
2
+ ẏ 2) dt
2
+ ẏ 2) dt
x
y
Curvature
)|
d2y
–––
dψ
ẋ ÿ – ẍ ẏ
dx2
–––––––––––––
κ = –––
= ––––––––
=
3
/
2
2
ds
dy 2 3/2
(ẋ + ẏ ) 2
1 + ––
dx
( [ ])
1
Radius of curvature ρ = ––
κ,
Centre of curvature c = r + ρ n^
L'Hôpital’s rule
If f(a) = g(a) = 0 and g'(a) ≠ 0 then
f(x)
f'(a)
Lim ––––
= ––––
x ➝a g(x)
g'(a)
Multi-variable calculus
∂g/∂x
∂w
∂w
∂w
For w = g(x, y, z), δw = ––– δx + ––– δy + ––– δz
grad g = ∂g/∂y
∂x
∂y
∂z
∂g/∂z
( )
CALCULUS
1
–––––––
√(x2 – 1)
1
–––––––
(1 – x2)
arcosh x
Integration f(x)
sec2 kx
tan x
cot x
cosec x
Centre of mass (uniform bodies)
Triangular lamina:
Solid hemisphere of radius r:
Hemispherical shell of radius r:
Solid cone or pyramid of height h:
Moments of inertia (uniform bodies, mass M)
2– along median from vertex
3
3– r from centre
8
1– r from centre
2
1– h above the base on the
4
line from centre of
base to vertex
Sector of circle, radius r, angle 2θ:
2r sin θ
–––––––
from centre
3θ
r sin θ
from centre
Arc of circle, radius r, angle 2θ at centre: –––––––
θ
1– h above the base on the
Conical shell, height h:
3 line from the centre of
Thin rod, length 2l, about perpendicular axis through centre:
Rectangular lamina about axis in plane bisecting edges of length 2l:
Thin rod, length 2l, about perpendicular axis through end:
Rectangular lamina about edge perpendicular to edges of length 2l:
Rectangular lamina, sides 2a and 2b, about perpendicular
1–
axis through centre:
M(a2 + b2)
3
Hoop or cylindrical shell of radius r about perpendicular
axis through centre:
Hoop of radius r about a diameter:
Disc or solid cylinder of radius r about axis:
base to the vertex
Solid sphere of radius r about a diameter:
5
Motion in polar coordinates
Spherical shell of radius r about a diameter:
Transverse velocity:
v = rθ̇
v2
Radial acceleration:
–rθ̇ 2 = – ––
r
Transverse acceleration: v̇ = rθ̈
General motion
Radial velocity:
Transverse velocity:
Radial acceleration:
Transverse acceleration:
ṙ
rθ̇
r̈ – rθ̇ 2
1 d
rθ̈ + 2ṙ θ̇ = – –– (r2θ̇ )
r dt
Moments as vectors
The moment about O of F acting at r is r × F
Mr2
1–
Mr2
2
1–
Mr2
2
1–
Mr2
4
2–
Mr2
5
2–
Mr2
3
Parallel axes theorem:
IA = IG + M(AG)2
Perpendicular axes theorem:
Iz = Ix + Iy (for a lamina in the (x, y) plane)
MECHANICS
Disc of radius r about a diameter:
Motion in a circle
1–
Ml2
3
1–
Ml2
3
4–
Ml2
3
4–
Ml2
3
Probability
Product-moment correlation: Pearson’s coefficient
∑ xi yi
–xy
Sxy
Σ( xi – x )( yi – y )
n
r=
=
=
2
2
Sxx Syy
 ∑ xi 2
  ∑ yi 2

Σ( xi – x ) Σ( yi – y )
− x2
− y2

 n

 n
P(A∪B) = P(A) + P(B) – P(A∩B)
P(A∩B) = P(A) . P(B|A)
P(B|A)P(A)
P(A|B) = ––––––––––––––––––––––
P(B|A)P(A) + P(B|A')P(A')
[
P(Aj)P(B|Aj)
Bayes’ Theorem: P(A j |B) = ––––––––––––
∑P(Ai)P(B|Ai)
Rank correlation: Spearman’s coefficient
Populations
6∑di2
rs = 1 – ––––––––
n(n2 – 1)
Discrete distributions
X is a random variable taking values xi in a discrete distribution with
P(X = xi) = pi
Expectation:
µ = E(X) = ∑xi pi
Variance:
σ2 = Var(X) = ∑(xi – µ)2 pi = ∑xi2pi – µ2
For a function g(X): E[g(X)] = ∑g(xi)pi
Regression
6
X is a continuous variable with probability density function (p.d.f.) f(x)
Expectation:
µ = E(X) = ∫ x f(x)dx
Variance:
σ2 = Var (X)
= ∫(x – µ)2 f(x)dx = ∫x2 f(x)dx – µ2
For a function g(X):
E[g(X)] = ∫g(x)f(x)dx
Cumulative
x
distribution function F(x) = P(X x) = ∫–∞f(t)dt
Sxx = ∑(xi – x )2 = ∑xi2 –
n
, Syy = ∑(yi – y )2 = ∑yi2 –
(∑xi)(∑yi)
Sxy = ∑(xi – x )(yi – y ) = ∑xi yi – –––––––––
n
Covariance
Sxy
= ∑( xi – x )( yi – y ) = ∑ xi yi – x y
––––
n
n
n
1 ∑(x – x )2f
S2 for population variance σ 2 where S2 = ––––
i
i
n–1
Probability generating functions
For a discrete distribution
For a sample of n pairs of observations (xi, yi)
(∑xi)2
–––––
Least squares regression line of y on x: y – y = b(x – x )
∑ xi yi
–xy
Sxy
∑(xi – x) (yi – y )
n
= –––––––––––––––
=
b = –––
Sxx
∑ xi 2
∑(xi – x )2
– x2
n
Estimates
Unbiased estimates from a single sample
σ2
X for population mean µ; Var X = ––
n
(∑yi)2
–––––
n
,
G(t) = E(tX)
E(X) = G'(1); Var(X) = G"(1) + µ – µ2
GX + Y (t) = GX (t) GY (t) for independent X, Y
Moment generating functions:
MX(θ) = E(eθX)
E(X) = M'(0) = µ;
E(Xn) = M(n)(0)
Var(X) = M"(0) – {M'(0)}2
MX + Y (θ) = MX (θ) MY (θ) for independent X, Y
STATISTICS
Continuous distributions
Correlation and regression
]
Markov Chains
pn + 1 = pnP
Long run proportion p = pP
Bivariate distributions
Covariance
Cov(X, Y) = E[(X – µX)(Y – µY)] = E(XY) – µXµY
Cov(X, Y)
Product-moment correlation coefficient
ρ = ––––––––
σX σY
Sum and difference
Var(aX ± bY) = a2Var(X) + b2Var(Y) ± 2ab Cov (X,Y)
If X, Y are independent: Var(aX ± bY) = a2Var(X) + b2Var(Y)
E(XY) = E(X) E(Y)
Coding
X = aX' + b
⇒ Cov(X, Y) = ac Cov(X', Y')
Y = cY' + d
}
One-factor model: xij = µ + αi + εij, where εij ~ N(0,σ2)
Ti2 T 2
SSB = ∑ni ( x i – x )2 = ∑ –––
ni – ––
n
i
i
2
––
SST = ∑ ∑ (xij – x )2 = ∑ ∑ xij2 – T
n
i j
i j
Yi
α + βxi + εi
RSS
∑(yi – a – bxi)2
α + βf(xi) + εi
α + βxi + γzi + εi
∑(yi – a –
εi ~ N(0, σ2)
No. of parameters, p
2
bf(xi))2
∑(yi – a – bxi –
2
czi)2
3
a, b, c are estimates for α, β, γ.
For the model Yi = α + βxi + εi,
Sxy
σ2
b = ––– , b ~ N β, –––
,
S
Sxx
xx
(
b−β
)
σ̂ 2 / Sxx
σ 2∑xi2
–––––––
a = y – b x , a ~ N α, n S
xx
(
RSS
σ^2 = n––––
–p
~ tn –2
)
(x0 – x )2
a + bx0 ~ N(α + βx0, σ2 1– + –––––––
n
Sxx
2
(Sxy)
RSS = Syy – ––––– = Syy (1 – r2)
Sxx
{
}
Randomised response technique
–ny – (1 – θ)
E(p^) = ––––––––––
(2θ – 1)
[(2θ – 1) p + (1 – θ)][θ – (2θ – 1)p]
Var(p^) = ––––––––––––––––––––––––––––––
2
n(2θ – 1)
Factorial design
Interaction between 1st and 2nd of 3 treatments
(–)
{
(Abc – abc) + (AbC – abC)
(ABc – aBc) + (ABC – aBC)
––––––––––––––––––––– – ––––––––––––––––––––––
2
2
}
Exponential smoothing
y^n+1 = α yn + α(1 – α)yn–1 + α(1 – α)2 yn–2 + ... + α(1 – α)n–1 y1
+ (1 – α)ny0
^y = y^ + α(y – y^ )
n+1
n
n
n
y^n+1 = α yn + (1 – α) y^n
STATISTICS
7
Analysis of variance
Regression
Description
Pearson’s product
moment
correlation test
r=
Distribution
∑ xi yi
–xy
n
2
 ∑ xi 2

2   ∑ yi
–
– y2
x





n
 n
t-test for the
difference in the
means of
2 samples
6∑di2
rs = 1 – –––––––
n(n2 – 1)
8
Normal test for
a mean
x–µ
σ/ n
N(0, 1)
t-test for a mean
x–µ
s/ n
tn – 1
χ2
test
t-test for
paired sample
Normal test for the
difference in the
means of 2 samples
with different
variances
Description
∑
(f
o
– fe )
( x1 – x2 ) – µ
s/ n
N(0, 1)
+ n2 – 2
1
(n1 – 1)s12 + (n2 – 1)s22
where s2 = –––––––––––––––––––––––
n1 + n2 – 2
See tables
Wilcoxon Rank-sum
(or Mann-Whitney)
2-Sample test
Samples size m, n: m n
Wilcoxon
W = sum of ranks of
sample size m
Mann-Whitney
T = W – 1–2 m(m + 1)
See tables
p –θ
Normal test on
binomial proportion
 θ (1 – θ ) 
 n 
χ2 test for variance
(n – 1)s 2
σ2
( x – y ) – ( µ1 – µ2 )
σ 12 σ 2 2
+
n1
n2
tn
A statistic T is calculated
from the ranked data.
χ 2v
t with (n – 1)
degrees of
freedom
( x – y ) – ( µ1 – µ2 )
1 1
+
s
n1 n2
Distribution
Wilcoxon single
sample test
2
fe
Test statistic
F-test on ratio of
two variances
s12 /σ12
–––––––
s22 /σ22
, s12 > s22
N(0, 1)
χ2n – 1
Fn
1
–1, n2 –1
STATISTICS: HYPOTHESIS TESTS
Spearman rank
correlation test
Test statistic
Name
Function
Mean
Variance
p.g.f. G(t) (discrete)
m.g.f. M(θ) (continuous)
P(X = r) = nCr qn–rpr ,
for r = 0, 1, ... ,n , 0 < p < 1, q = 1 – p
np
npq
G(t) = (q + pt)n
Poisson (λ)
Discrete
λr
P(X = r) = e–λ –––
r! ,
for r = 0, 1, ... , λ > 0
λ
λ
G(t) = eλ(t – 1)
µ
σ2
M(θ) = exp(µθ + Wσ 2θ 2)
1
x–µ
exp – W –––––
σ
( (
Normal N(µ, σ2)
Continuous
f(x) =
Uniform (Rectangular) on
[a, b] Continuous
1
f(x) = –––––
b–a
Exponential
Continuous
f(x) = λe–λx
Geometric
Discrete
P(X = r) = q r – 1p ,
σ 2π
) ),
2
–∞ < x < ∞
9
Negative binomial
Discrete
, axb
,
0<p<1
x 0, λ > 0
r = 1, 2, ... ,
, q=1–p
P(X = r) = r – 1Cn – 1 qr – n pn ,
r = n, n + 1, ... ,
0<p<1 ,
q=1–p
a+b
–––––
2
1
––
12
(b – a)2
ebθ – eaθ
M(θ) = –––––––––
(b – a)θ
1
––
λ
1
––
λ2
λ
M(θ) = –––––
λ–θ
1
––
p
q
––2
p
pt
G(t) = –––––
1 – qt
n
––
p
nq
––2
p
pt
G(t) = –––––
1 – qt
(
n
)
STATISTICS: DISTRIBUTIONS
Binomial B(n, p)
Discrete
Numerical Solution of Equations
f(xn)
The Newton-Raphson iteration for solving f(x) = 0 : xn + 1 = xn – ––––
f'(xn)
Numerical integration
The trapezium rule
b
1
b–a
ydx ≈ –2 h{(y0 + yn) + 2(y1 + y2 + ... + yn–1)}, where h = –––––
n
a
∫
The mid-ordinate rule
∫a ydx ≈ h(y
b
1–
2
h2
f(a + h) = f(a) + hf '(a) + ––– f"(a + ξ), 0 < ξ < h
2!
f(x)
(x – a)2
= f(a) + (x – a)f '(a) + –––––– f"(a) + error
2!
f(x)
(x – a)2
= f(a) + (x – a)f '(a) + –––––– f"(η) , a < η < x
2!
b–a
+ y1 1– + ... + yn– 1 1– + yn– 1– ), where h = –––––
n
2
2
2
∫a ydx ≈ 1–3 h{(y0 + yn) + 4(y1 + y3 + ... + yn–1) + 2(y2 + y4 + ... + yn–2)},
b
b–a
where h = –––––
n
The Gaussian 2-point integration rule
10
  −h   h  
f ( x )dx ≈ h f   + f   
–h
  3  3
Interpolation/finite differences
h
Numerical solution of differential equations
dy
For –– = f(x, y):
dx
Euler’s method : yr + 1 = yr + hf(xr, yr); xr+1 = xr + h
Runge-Kutta method (order 2) (modified Euler method)
yr + 1 = yr + 1–2 (k1 + k2)
where k1 = h f(xr, yr), k2 = h f(xr + h, yr + k1)
Runge-Kutta method, order 4:
n
x – xi
Lagrange’s polynomial : Pn(x) = ∑ Lr(x)f(x) where Lr(x) = ∏ ––––––
x
i=0
r – xi
i≠r
Newton’s forward difference interpolation formula
(x – x0)(x – x1)
(x – x0)
––––––––––––
∆f(x
)
+
∆2f(x0) + ...
f(x) = f(x0) + ––––––
0
2!h2
h
Newton’s divided difference interpolation formula
yr+1 = yr + 1–6 (k1 + 2k2 + 2k3 + k4),
where k1 = hf(xr, yr)
k3 = hf(xr + 1–2 h, yr + 1–2 k2)
k2 = hf(xr + 1–2 h, yr + 1–2 k1)
k4 = hf(xr + h, yr + k3).
Logic gates
f(x) = f[x0] + (x – x0]f[x0, x1] + (x – x0) (x – x1)f[x0, x1, x2] + ...
Numerical differentiation
f(x + h) – 2f(x) + f(x – h)
f"(x) ≈ –––––––––––––––––––––
h2
NOT
AND
OR
NAND
NUMERICAL ANALYSIS
DECISION & DISCRETE MATHEMATICS
Simpson’s rule
for n even
∫
Taylor polynomials
h2
f(a + h) = f(a) + hf '(a) + ––– f"(a) + error
2!
Statistical Tables
12 – 17
18 – 20
21
22
23
23
24 – 25
26 – 27
28 – 29
30
30
31
31 – 32
Cumulative binomial probability
Cumulative Poisson probability
Critical values for correlation coefficients
The Normal distribution and its inverse
Percentage points of the χ2 distribution
Percentage points of the t-distribution
Critical values for the F-test
Critical values for the Mann-Whitney test
Critical values for the Wilcoxon Rank Sum 2-sample test
Critical values for the Wilcoxon Single sample and Paired sample tests
Shewhart Chart: Action and Warning lines
Estimation of standard deviation from range
Random permutations
11
The Binomial distribution: cumulative probabilites
x
P (X x) = ∑ nCr (1 – p) n–r pr
r=0
n
p
0.250 0.300
1/3
0.350 0.400 0.450
0.500 0.550
0.600 0.650 2/3
2
0
1
2
0.9025 0.8100 0.7225 0.6944 0.6400 0.5625 0.4900 0.4444 0.4225 0.3600 0.3025 0.2500 0.2025 0.1600 0.1225 0.1111 0.0900 0.0625 0.0400 0.0278 0.0225 0.0100 0.0025
0.9975 0.9900 0.9775 0.9722 0.9600 0.9375 0.9100 0.8889 0.8775 0.8400 0.7975 0.7500 0.6975 0.6400 0.5775 0.5556 0.5100 0.4375 0.3600 0.3056 0.2775 0.1900 0.0975
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3
0
1
2
3
0.8574
0.9928
0.9999
1.0000
0.7290
0.9720
0.9990
1.0000
0.6141
0.9392
0.9966
1.0000
0.5787
0.9259
0.9954
1.0000
0.5120
0.8960
0.9920
1.0000
0.4219
0.8437
0.9844
1.0000
0.3430
0.7840
0.9730
1.0000
0.2963
0.7407
0.9630
1.0000
0.2746
0.7183
0.9571
1.0000
0.2160
0.6480
0.9360
1.0000
0.1664
0.5748
0.9089
1.0000
0.1250
0.5000
0.8750
1.0000
0.0911
0.4252
0.8336
1.0000
0.0640
0.3520
0.7840
1.0000
0.0429
0.2818
0.7254
1.0000
0.0370
0.2593
0.7037
1.0000
0.0270
0.2160
0.6570
1.0000
0.0156
0.1563
0.5781
1.0000
0.0080
0.1040
0.4880
1.0000
0.0046
0.0741
0.4213
1.0000
0.0034
0.0608
0.3859
1.0000
0.0010
0.0280
0.2710
1.0000
0.0001
0.0073
0.1426
1.0000
4
0
1
2
3
4
0.8145
0.9860
0.9995
1.0000
0.6561
0.9477
0.9963
0.9999
1.0000
0.5220
0.8905
0.9880
0.9995
1.0000
0.4823
0.8681
0.9838
0.9992
1.0000
0.4096
0.8192
0.9728
0.9984
1.0000
0.3164
0.7383
0.9492
0.9961
1.0000
0.2401
0.6517
0.9163
0.9919
1.0000
0.1975
0.5926
0.8889
0.9877
1.0000
0.1785
0.5630
0.8735
0.9850
1.0000
0.1296
0.4752
0.8208
0.9744
1.0000
0.0915
0.3910
0.7585
0.9590
1.0000
0.0625
0.3125
0.6875
0.9375
1.0000
0.0410
0.2415
0.6090
0.9085
1.0000
0.0256
0.1792
0.5248
0.8704
1.0000
0.0150
0.1265
0.4370
0.8215
1.0000
0.0123
0.1111
0.4074
0.8025
1.0000
0.0081
0.0837
0.3483
0.7599
1.0000
0.0039
0.0508
0.2617
0.6836
1.0000
0.0016
0.0272
0.1808
0.5904
1.0000
0.0008
0.0162
0.1319
0.5177
1.0000
0.0005
0.0120
0.1095
0.4780
1.0000
0.0001
0.0037
0.0523
0.3439
1.0000
0.0000
0.0005
0.0140
0.1855
1.0000
5
0
1
2
3
4
5
0.7738
0.9774
0.9988
1.0000
0.5905
0.9185
0.9914
0.9995
1.0000
0.4437
0.8352
0.9734
0.9978
0.9999
1.0000
0.4019
0.8038
0.9645
0.9967
0.9999
1.0000
0.3277
0.7373
0.9421
0.9933
0.9997
1.0000
0.2373
0.6328
0.8965
0.9844
0.9990
1.0000
0.1681
0.5282
0.8369
0.9692
0.9976
1.0000
0.1317
0.4609
0.7901
0.9547
0.9959
1.0000
0.1160
0.4284
0.7648
0.9460
0.9947
1.0000
0.0778
0.3370
0.6826
0.9130
0.9898
1.0000
0.0503
0.2562
0.5931
0.8688
0.9815
1.0000
0.0313
0.1875
0.5000
0.8125
0.9688
1.0000
0.0185
0.1312
0.4069
0.7438
0.9497
1.0000
0.0102
0.0870
0.3174
0.6630
0.9222
1.0000
0.0053
0.0540
0.2352
0.5716
0.8840
1.0000
0.0041
0.0453
0.2099
0.5391
0.8683
1.0000
0.0024
0.0308
0.1631
0.4718
0.8319
1.0000
0.0010
0.0156
0.1035
0.3672
0.7627
1.0000
0.0003
0.0067
0.0579
0.2627
0.6723
1.0000
0.0001
0.0033
0.0355
0.1962
0.5981
1.0000
0.0001
0.0022
0.0266
0.1648
0.5563
1.0000
0.0000
0.0005
0.0086
0.0815
0.4095
1.0000
0.0000
0.0012
0.0226
0.2262
1.0000
0
1
2
3
4
5
6
0.7351
0.9672
0.9978
0.9999
1.0000
0.5314
0.8857
0.9841
0.9987
0.9999
1.0000
0.3771
0.7765
0.9527
0.9941
0.9996
1.0000
0.3349
0.7368
0.9377
0.9913
0.9993
1.0000
0.2621
0.6554
0.9011
0.9830
0.9984
0.9999
1.0000
0.1780
0.5339
0.8306
0.9624
0.9954
0.9998
1.0000
0.1176
0.4202
0.7443
0.9295
0.9891
0.9993
1.0000
0.0878
0.3512
0.6804
0.8999
0.9822
0.9986
1.0000
0.0754
0.3191
0.6471
0.8826
0.9777
0.9982
1.0000
0.0467
0.2333
0.5443
0.8208
0.9590
0.9959
1.0000
0.0277
0.1636
0.4415
0.7447
0.9308
0.9917
1.0000
0.0156
0.1094
0.3438
0.6563
0.8906
0.9844
1.0000
0.0083
0.0692
0.2553
0.5585
0.8364
0.9723
1.0000
0.0041
0.0410
0.1792
0.4557
0.7667
0.9533
1.0000
0.0018
0.0223
0.1174
0.3529
0.6809
0.9246
1.0000
0.0014
0.0178
0.1001
0.3196
0.6488
0.9122
1.0000
0.0007
0.0109
0.0705
0.2557
0.5798
0.8824
1.0000
0.0002
0.0046
0.0376
0.1694
0.4661
0.8220
1.0000
0.0001
0.0016
0.0170
0.0989
0.3446
0.7379
1.0000
0.0000
0.0007
0.0087
0.0623
0.2632
0.6651
1.0000
0.0000
0.0004
0.0059
0.0473
0.2235
0.6229
1.0000
0.0000
0.0001
0.0013
0.0159
0.1143
0.4686
1.0000
0.0000
0.0001
0.0022
0.0328
0.2649
1.0000
0
1
2
3
4
5
6
7
0.6983
0.9556
0.9962
0.9998
1.0000
0.4783
0.8503
0.9743
0.9973
0.9998
1.0000
0.3206
0.7166
0.9262
0.9879
0.9988
0.9999
1.0000
0.2791
0.6698
0.9042
0.9824
0.9980
0.9999
1.0000
1.0000
0.2097
0.5767
0.8520
0.9667
0.9953
0.9996
1.0000
1.0000
0.1335
0.4449
0.7564
0.9294
0.9871
0.9987
0.9999
1.0000
0.0824
0.3294
0.6471
0.8740
0.9712
0.9962
0.9998
1.0000
0.0585
0.2634
0.5706
0.8267
0.9547
0.9931
0.9995
1.0000
0.0490
0.2338
0.5323
0.8002
0.9444
0.9910
0.9994
1.0000
0.0280
0.1586
0.4199
0.7102
0.9037
0.9812
0.9984
1.0000
0.0152
0.1024
0.3164
0.6083
0.8471
0.9643
0.9963
1.0000
0.0078
0.0625
0.2266
0.5000
0.7734
0.9375
0.9922
1.0000
0.0037
0.0357
0.1529
0.3917
0.6836
0.8976
0.9848
1.0000
0.0016
0.0188
0.0963
0.2898
0.5801
0.8414
0.9720
1.0000
0.0006
0.0090
0.0556
0.1998
0.4677
0.7662
0.9510
1.0000
0.0005
0.0069
0.0453
0.1733
0.4294
0.7366
0.9415
1.0000
0.0002
0.0038
0.0288
0.1260
0.3529
0.6706
0.9176
1.0000
0.0001
0.0013
0.0129
0.0706
0.2436
0.5551
0.8665
1.0000
0.0000
0.0004
0.0047
0.0333
0.1480
0.4233
0.7903
1.0000
0.0000
0.0001
0.0020
0.0176
0.0958
0.3302
0.7209
1.0000
0.0000
0.0001
0.0012
0.0121
0.0738
0.2834
0.6794
1.0000
0.0000
0.0002
0.0027
0.0257
0.1497
0.5217
1.0000
0.0000
0.0002
0.0038
0.0444
0.3017
1.0000
CUMULATIVE BINOMIAL PROBABILITY
0.9500 0.9000 0.8500 0.8333 0.8000 0.7500 0.7000 0.6667 0.6500 0.6000 0.5500 0.5000 0.4500 0.4000 0.3500 0.3333 0.3000 0.2500 0.2000 0.1667 0.1500 0.1000 0.0500
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
12
0
1
7
0.200
0.850 0.900 0.950
0.050 0.100
1
6
0.150 1/6
0.700 0.750 0.800 5/6
x
n
8
9
13
11
0.050 0.100
0.150 1/6
0.200
0.250 0.300
1/3
0.350 0.400 0.450
0.500 0.550
0.600 0.650 2/3
0.700 0.750 0.800
5/6
0.850 0.900 0.950
0
1
2
3
4
5
6
7
8
0.6634
0.9428
0.9942
0.9996
1.0000
0.4305
0.8131
0.9619
0.9950
0.9996
1.0000
0.2725
0.6572
0.8948
0.9786
0.9971
0.9998
1.0000
0.2326
0.6047
0.8652
0.9693
0.9954
0.9996
1.0000
0.1678
0.5033
0.7969
0.9437
0.9896
0.9988
0.9999
1.0000
0.1001
0.3671
0.6785
0.8862
0.9727
0.9958
0.9996
1.0000
0.0576
0.2553
0.5518
0.8059
0.9420
0.9887
0.9987
0.9999
1.0000
0.0390
0.1951
0.4682
0.7414
0.9121
0.9803
0.9974
0.9998
1.0000
0.0319
0.1691
0.4278
0.7064
0.8939
0.9747
0.9964
0.9998
1.0000
0.0168
0.1064
0.3154
0.5941
0.8263
0.9502
0.9915
0.9993
1.0000
0.0084
0.0632
0.2201
0.4770
0.7396
0.9115
0.9819
0.9983
1.0000
0.0039
0.0352
0.1445
0.3633
0.6367
0.8555
0.9648
0.9961
1.0000
0.0017
0.0181
0.0885
0.2604
0.5230
0.7799
0.9368
0.9916
1.0000
0.0007
0.0085
0.0498
0.1737
0.4059
0.6846
0.8936
0.9832
1.0000
0.0002
0.0036
0.0253
0.1061
0.2936
0.5722
0.8309
0.9681
1.0000
0.0002
0.0026
0.0197
0.0879
0.2587
0.5318
0.8049
0.9610
1.0000
0.0001
0.0013
0.0113
0.0580
0.1941
0.4482
0.7447
0.9424
1.0000
0.0000
0.0004
0.0042
0.0273
0.1138
0.3215
0.6329
0.8999
1.0000
0.0000
0.0001
0.0012
0.0104
0.0563
0.2031
0.4967
0.8322
1.0000
0.0000
0.0004
0.0046
0.0307
0.1348
0.3953
0.7674
1.0000
0.0000
0.0002
0.0029
0.0214
0.1052
0.3428
0.7275
1.0000
0.0000
0.0004
0.0050
0.0381
0.1869
0.5695
1.0000
0.0000
0.0004
0.0058
0.0572
0.3366
1.0000
0
1
2
3
4
5
6
7
8
9
0.6302
0.9288
0.9916
0.9994
1.0000
0.3874
0.7748
0.9470
0.9917
0.9991
0.9999
1.0000
0.2316
0.5995
0.8591
0.9661
0.9944
0.9994
1.0000
0.1938
0.5427
0.8217
0.9520
0.9911
0.9989
0.9999
1.0000
0.1342
0.4362
0.7382
0.9144
0.9804
0.9969
0.9997
1.0000
0.0751
0.3003
0.6007
0.8343
0.9511
0.9900
0.9987
0.9999
1.0000
0.0404
0.1960
0.4628
0.7297
0.9012
0.9747
0.9957
0.9996
1.0000
0.0260
0.1431
0.3772
0.6503
0.8552
0.9576
0.9917
0.9990
0.9999
1.0000
0.0207
0.1211
0.3373
0.6089
0.8283
0.9464
0.9888
0.9986
0.9999
1.0000
0.0101
0.0705
0.2318
0.4826
0.7334
0.9006
0.9750
0.9962
0.9997
1.0000
0.0046
0.0385
0.1495
0.3614
0.6214
0.8342
0.9502
0.9909
0.9992
1.0000
0.0020
0.0195
0.0898
0.2539
0.5000
0.7461
0.9102
0.9805
0.9980
1.0000
0.0008
0.0091
0.0498
0.1658
0.3786
0.6386
0.8505
0.9615
09954
1.0000
0.0003
0.0038
0.0250
0.0994
0.2666
0.5174
0.7682
0.9295
0.9899
1.0000
0.0001
0.0014
0.0112
0.0536
0.1717
0.3911
0.6627
0.8789
0.9793
1.0000
0.0001
0.0010
0.0083
0.0424
0.1448
0.3497
0.6228
0.8569
0.9740
1.0000
0.0000
0.0004
0.0043
0.0253
0.0988
0.2703
0.5372
0.8040
0.9596
1.0000
00000
0.0001
0.0013
0.0100
0.0489
0.1657
0.3993
0.6997
0.9249
1.0000
0.0000
0.0003
0.0031
0.0196
0.0856
0.2618
0.5638
0.8658
1.0000
0.0000
0.0001
0.0011
0.0090
0.0480
0.1783
0.4573
0.8062
1.0000
0.0000
0.0006
0.0056
0.0339
0.1409
0.4005
0.7684
1.0000
0.0000
0.0001
0.0009
0.0083
0.0530
0.2252
0.6126
1.0000
0.0000
0.0006
0.0084
0.0712
0.3698
1.0000
0
1
2
3
4
5
6
7
8
9
10
0.5987
0.9139
0.9885
0.9990
0.9999
1.0000
0.3487
0.7361
0.9298
0.9872
0.9984
0.9999
1.0000
0.1969
0.5443
0.8202
0.9500
0.9901
0.9986
0.9999
1.0000
0.1615
0.4845
0.7752
0.9303
0.9845
0.9976
0.9997
1.0000
0.1074
0.3758
0.6778
0.8791
0.9672
0.9936
0.9991
0.9999
1.0000
0.0563
0.2440
0.5256
0.7759
0.9219
0.9803
0.9965
0.9996
1.0000
0.0282
0.1493
0.3828
0.6496
0.8497
0.9527
0.9894
0.9984
0.9999
1.0000
0.0173
0.1040
0.2991
0.5593
0.7869
0.9234
0.9803
0.9966
0.9996
1.0000
0.0135
0.0860
0.2616
0.5138
0.7515
0.9051
0.9740
0.9952
0.9995
1.0000
0.0060
0.0464
0.1673
0.3823
0.6331
0.8338
0.9452
0.9877
0.9983
0.9999
1.0000
0.0025
0.0233
0.0996
0.2660
0.5044
0.7384
0.8980
0.9726
0.9955
0.9997
1.0000
0.0010
0.0107
0.0547
0.1719
0.3770
0.6230
0.8281
0.9453
0.9893
0.9990
1.0000
0.0003
0.0045
0.0274
0.1020
0.2616
0.4956
0.7340
0.9004
0.9767
0.9975
1.0000
0.0001
0.0017
0.0123
0.0548
0.1662
0.3669
0.6177
0.8327
0.9536
0.9940
1.0000
0.0000
0.0005
0.0048
0.0260
0.0949
0.2485
0.4862
0.7384
0.9140
0.9865
1.0000
0.0000
0.0004
0.0034
0.0197
0.0766
0.2131
0.4407
0.7009
0.8960
0.9827
1.0000
0.0000
0.0001
0.0016
0.0106
0.0473
0.1503
0.3504
0.6172
0.8507
0.9718
1.0000
0.0000
0.0004
0.0035
0.0197
0.0781
0.2241
0.4744
0.7560
0.9437
1.0000
0.0000
0.0001
0.0009
0.0064
0.0328
0.1209
0.3222
0.6242
0.8926
1.0000
0.0000
0.0003
0.0024
0.0155
0.0697
0.2248
0.5155
0.8385
1.0000
0.0000
0.0001
0.0014
0.0099
0.0500
0.1798
0.4557
0.8031
1.0000
0.0000
0.0001
0.0016
0.0128
0.0702
0.2639
0.6513
1.0000
0.0000
0.0001
0.0010
0.0115
0.0861
0.4013
1.0000
0
1
2
3
4
5
6
7
8
9
10
11
0.5688
0.8981
0.9848
0.9984
0.9999
1.0000
0.3138
0.6974
0.9104
0.9815
0.9972
0.9997
1.0000
0.1673
0.4922
0.7788
0.9306
0.9841
0.9973
0.9997
1.0000
0.1346
0.4307
0.7268
0.9044
0.9755
0.9954
0.9994
0.9999
1.0000
0.0859
0.3221
0.6174
0.8389
0.9496
0.9883
0.9980
0.9998
1.0000
0.0422
0.1971
0.4552
0.7133
0.8854
0.9657
0.9924
0.9988
0.9999
1.0000
0.0198
0.1130
0.3127
0.5696
0.7897
0.9218
0.9784
0.9957
0.9994
1.0000
0.0116
0.0751
0.2341
0.4726
0.7110
0.8779
0.9614
0.9912
0.9986
0.9999
1.0000
0.0088
0.0606
0.2001
0.4256
0.6683
0.8513
0.9499
0.9878
0.9980
0.9998
1.0000
0.0036
0.0302
0.1189
0.2963
0.5328
0.7535
0.9006
0.9707
0.9941
0.9993
1.0000
0.0014
0.0139
0.0652
0.1911
0.3971
0.6331
0.8262
0.9390
0.9852
0.9978
0.9998
1.0000
0.0005
0.0059
0.0327
0.1133
0.2744
0.5000
0.7256
0.8867
0.9673
0.9941
0.9995
1.0000
0.0002
0.0022
0.0148
0.0610
0.1738
0.3669
0.6029
0.8089
0.9348
0.9861
0.9986
1.0000
0.0000
0.0007
0.0059
0.0293
0.0994
0.2465
0.4672
0.7037
0.8811
0.9698
0.9964
1.0000
0.0000
0.0002
0.0020
0.0122
0.0501
0.1487
0.3317
0.5744
0.7999
0.9394
0.9912
1.0000
.00000
0.0001
0.0014
0.0088
0.0386
0.1221
0.2890
0.5274
0.7659
0.9249
0.9884
1.0000
0.0000
0.0006
0.0043
0.0216
0.0782
0.2103
0.4304
0.6873
0.8870
0.9802
1.0000
0.0000
0.0001
0.0012
0.0076
0.0343
0.1146
0.2867
0.5448
0.8029
0.9578
1.0000
0.0000
0.0002
0.0020
0.0117
0.0504
0.1611
0.3826
0.6779
0.9141
1.0000
0.0000
0.0001
0.0006
0.0046
0.0245
0.0956
0.2732
0.5693
0.8654
1.0000
0.0000
0.0003
0.0027
0.0159
0.0694
0.2212
0.5078
0.8327
1.0000
0.0000
0.0003
0.0028
0.0185
0.0896
0.3026
0.6862
1.0000
0.0000
0.0001
0.0016
0.0152
0.1019
0.4312
1.0000
CUMULATIVE BINOMIAL PROBABILITY
10
p
x
n
12
14
14
0.050 0.100
0.150 1/6
0.200
0.250 0.300
1/3
0.350 0.400 0.450
0.500 0.550
0.600 0.650 2/3
0.700 0.750 0.800
5/6
0.850 0.900 0.950
0
1
2
3
4
5
6
7
8
9
10
11
12
0.5404
0.8816
0.9804
0.9978
0.9998
1.0000
0.2824
0.6590
0.8891
0.9744
0.9957
0.9995
0.9999
1.0000
0.1422
0.4435
0.7358
0.9078
0.9761
0.9954
0.9993
0.9999
1.0000
0.1122
0.3813
0.6774
0.8748
0.9637
0.9921
0.9987
0.9998
1.0000
0.0687
0.2749
0.5583
0.7946
0.9274
0.9806
0.9961
0.9994
0.9999
1.0000
0.0317
0.1584
0.3907
0.6488
0.8424
0.9456
0.9857
0.9972
0.9996
1.0000
0.0138
0.0850
0.2528
0.4925
0.7237
0.8822
0.9614
0.9905
0.9983
0.9998
1.0000
0.0077
0.0540
0.1811
0.3931
0.6315
0.8223
0.9336
0.9812
0.9961
0.9995
1.0000
0.0057
0.0424
0.1513
0.3467
0.5833
0.7873
0.9154
0.9745
0.9944
0.9992
0.9999
1.0000
0.0022
0.0196
0.0834
0.2253
0.4382
0.6652
0.8418
0.9427
0.9847
0.9972
0.9997
1.0000
0.0008
0.0083
0.0421
0.1345
0.3044
0.5269
0.7393
0.8883
0.9644
0.9921
0.9989
0.9999
1.0000
0.0002
0.0032
0.0193
0.0730
0.1938
0.3872
0.6128
0.8062
0.9270
0.9807
0.9968
0.9998
1.0000
0.0001
0.0011
0.0079
0.0356
0.1117
0.2607
0.4731
0.6956
0.8655
0.9579
0.9917
0.9992
1.0000
0.0000
0.0003
0.0028
0.0153
0.0573
0.1582
0.3348
0.5618
0.7747
0.9166
0.9804
0.9978
1.0000
0.0000
0.0001
0.0008
0.0056
0.0255
0.0846
0.2127
0.4167
0.6533
0.8487
0.9576
0.9943
1.0000
0.0000
0.0005
0.0039
0.0188
0.0664
0.1777
0.3685
0.6069
0.8189
0.9460
0.9923
1.0000
0.0000
0.0002
0.0017
0.0095
0.0386
0.1178
0.2763
0.5075
0.7472
0.9150
0.9862
1.0000
0.0000
0.0004
0.0028
0.0143
0.0544
0.1576
0.3512
0.6093
0.8416
0.9683
1.0000
0.0000
0.0001
0.0006
0.0039
0.0194
0.0726
0.2054
0.4417
0.7251
0.9313
1.0000
0.0000
0.0002
0.0013
0.0079
0.0364
0.1252
0.3226
0.6187
0.8878
1.0000
0.0000
0.0001
0.0007
0.0046
0.0239
0.0922
0.2642
0.5565
0.8578
1.0000
0.0000
0.0001
0.0005
0.0043
0.0256
0.1109
0.3410
0.7176
1.0000
0.0000
0.0002
0.0022
0.0196
0.1184
0.4596
1.0000
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0.5133
0.8646
0.9755
0.9969
0.9997
1.0000
0.2542
0.6213
0.8661
0.9658
0.9935
0.9991
0.9999
1.0000
01209
0.3983
0.6920
0.8820
0.9658
0.9925
0.9987
0.9998
1.0000
0.0935
0.3365
0.6281
0.8419
0.9488
0.9873
0.9976
0.9997
1.0000
0.0550
0.2336
0.5017
0.7473
0.9009
0.9700
0.9930
0.9988
0.9998
1.0000
0.0238
0.1267
0.3326
0.5843
0.7940
0.9198
0.9757
0.9944
0.9990
0.9999
1.0000
0.0097
0.0637
0.2025
0.4206
0.6543
0.8346
0.9376
0.9818
0.9960
0.9993
0.9999
1.0000
0.0051
0.0385
0.1387
0.3224
0.5520
0.7587
0.8965
0.9653
0.9912
0.9984
0.9998
1.0000
0.0037
0.0296
0.1132
0.2783
0.5005
0.7159
0.8705
0.9538
0.9874
0.9975
0.9997
1.0000
0.0013
0.0126
0.0579
0.1686
0.3530
0.5744
0.7712
0.9023
0.9679
0.9922
0.9987
0.9999
1.0000
0.0004
0.0049
0.0269
0.0929
0.2279
0.4268
0.6437
0.8212
0.9302
0.9797
0.9959
0.9995
1.0000
0.0001
0.0017
0.0112
0.0461
0.1334
0.2905
0.5000
0.7095
0.8666
0.9539
0.9888
0.9983
0.9999
1.0000
0.0000
0.0005
0.0041
0.0203
0.0698
0.1788
0.3563
0.5732
0.7721
0.9071
0.9731
0.9951
0.9996
1.0000
0.0000
0.0001
0.0013
0.0078
0.0321
0.0977
0.2288
0.4256
0.6470
0.8314
0.9421
0.9874
0.9987
1.0000
0.0000
0.0003
0.0025
0.0126
0.0462
0.1295
0.2841
0.4995
0.7217
0.8868
0.9704
0.9963
1.0000
0.0000
0.0002
0.0016
0.0088
0.0347
0.1035
0.2413
0.4480
0.6776
0.8613
0.9615
0.9949
1.0000
0.0000
0.0001
0.0007
0.0040
0.0182
0.0624
0.1654
0.3457
0.5794
0.7975
0.9363
0.9903
1.0000
0.0000
0.0001
0.0010
0.0056
0.0243
0.0802
0.2060
0.4157
0.6674
0.8733
0.9762
1.0000
0.0000
0.0002
0.0012
0.0070
0.0300
0.0991
0.2527
0.4983
0.7664
0.9450
1.0000
0.0000
0.0003
0.0024
0.0127
0.0512
0.1581
0.3719
0.6635
0.9065
1.0000
0.0000
0.0002
0.0013
0.0075
0.0342
0.1180
0.3080
0.6017
0.8791
1.0000
0.0000
0.0001
0.0009
0.0065
0.0342
0.1339
0.3787
0.7458
1.0000
0.0000
0.0003
0.0031
0.0245
0.1354
0.4867
1.0000
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.4877
0.8470
0.9699
0.9958
0.9996
1.0000
0.2288
0.5846
0.8416
0.9559
0.9908
0.9985
0.9998
1.0000
0.1028
0.3567
0.6479
0.8535
0.9533
0.9885
0.9978
0.9997
1.0000
0.0779
0.2960
0.5795
0.8063
0.9310
0.9809
0.9959
0.9993
0.9999
1.0000
0.0440
0.1979
0.4481
0.6982
0.8702
0.9561
0.9884
0.9976
0.9996
1.0000
0.0178
0.1010
0.2811
0.5213
0.7415
0.8883
0.9617
0.9897
0.9978
0.9997
1.0000
0.0068
0.0475
0.1608
0.3552
0.5842
0.7805
0.9067
0.9685
0.9917
0.9983
0.9998
1.0000
0.0034
0.0274
0.1053
0.2612
0.4755
0.6898
0.8505
0.9424
0.9826
0.9960
0.9993
0.9999
1.0000
0.0024
0.0205
0.0839
0.2205
0.4227
0.6405
0.8164
0.9247
0.9757
0.9940
0.9989
0.9999
1.0000
0.0008
0.0081
0.0398
0.1243
0.2793
0.4859
0.6925
0.8499
0.9417
0.9825
0.9961
0.9994
0.9999
1.0000
0.0002
0.0029
0.0170
0.0632
0.1672
0.3373
0.5461
0.7414
0.8811
0.9574
0.9886
0.9978
0.9997
1.0000
0.0001
0.0009
0.0065
0.0287
0.0898
0.2120
0.3953
0.6047
0.7880
0.9102
0.9713
0.9935
0.9991
0.9999
1.0000
0.0000
0.0003
0.0022
0.0114
0.0426
0.1189
0.2586
0.4539
0.6627
0.8328
0.9368
0.9830
0.9971
0.9998
1.0000
0.0000
0.0001
0.0006
0.0039
0.0175
0.0583
0.1501
0.3075
0.5141
0.7207
0.8757
0.9602
0.9919
0.9992
1.0000
0.0000
0.0001
0.0011
0.0060
0.0243
0.0753
0.1836
0.3595
0.5773
0.7795
0.9161
0.9795
0.9976
1.0000
0.0000
0.0001
0.0007
0.0040
0.0174
0.0576
0.1495
0.3102
0.5245
0.7388
0.8947
0.9726
0.9966
1.0000
0.0000
0.0002
0.0017
0.0083
0.0315
0.0933
0.2195
0.4158
0.6448
0.8392
0.9525
0.9932
1.0000
0.0000
0.0003
0.0022
0.0103
0.0383
0.1117
0.2585
0.4787
0.7189
0.8990
0.9822
1.0000
0.0000
0.0004
0.0024
0.0116
0.0439
0.1298
0.3018
0.5519
0.8021
0.9560
1.0000
0.0000
0.0001
0.0007
0.0041
0.0191
0.0690
0.1937
0.4205
0.7040
0.9221
1.0000
0.0000
0.0003
0.0022
0.0115
0.0467
0.1465
0.3521
0.6433
0.8972
1.0000
0.0000
0.0002
0.0015
0.0092
0.0441
0.1584
0.4154
0.7712
1.0000
0.0000
0.0004
0.0042
0.0301
0.1530
0.5123
1.0000
CUMULATIVE BINOMIAL PROBABILITY
13
p
x
n
15
p
x
0.050
0.4633
0.8290
0.9638
0.9945
0.9994
0.9999
1.0000
0.4401
0.8108
0.9571
0.9930
0.9991
0.9999
1.0000
0.4181
0.7922
0.9497
0.9912
0.9988
0.9999
1.0000
0.100
0.2059
0.5490
0.8159
0.9444
0.9873
0.9978
0.9997
1.0000
0.1853
0.5147
0.7892
0.9316
0.9830
0.9967
0.9995
0.9999
1.0000
0.1668
0.4818
0.7618
0.9174
0.9779
0.9953
0.9992
0.9999
1.0000
0.150
0.0874
0.3186
0.6042
0.8227
0.9383
0.9832
0.9964
0.9994
0.9999
1.0000
1/6
0.0649
0.2596
0.5322
0.7685
0.9102
0.9726
0.9934
0.9987
0.9998
1.0000
0.200
0.0352
0.1671
0.3980
0.6482
0.8358
0.9389
0.9819
0.9958
0.9992
0.9999
1.0000
0.250
0.0134
0.0802
0.2361
0.4613
0.6865
0.8516
0.9434
0.9827
0.9958
0.9992
0.9999
1.0000
0.300
0.0047
0.0353
0.1268
0.2969
0.5155
0.7216
0.8689
0.9500
0.9848
0.9963
0.9993
0.9999
1.0000
1/3
0.0023
0.0194
0.0794
0.2092
0.4041
0.6184
0.7970
0.9118
0.9692
0.9915
0.9982
0.9997
1.0000
0.350
0.0016
0.0142
0.0617
0.1727
0.3519
0.5643
0.7548
0.8868
0.9578
0.9876
0.9972
0.9995
0.9999
1.0000
0.400
0.0005
0.0052
0.0271
0.0905
0.2173
0.4032
0.6098
0.7869
0.9050
0.9662
0.9907
0.9981
0.9997
1.0000
0.450
0.0001
0.0017
0.0107
0.0424
0.1204
0.2608
0.4522
0.6535
0.8182
0.9231
0.9745
0.9937
0.9989
0.9999
1.0000
0.500
0.0000
0.0005
0.0037
0.0176
0.0592
0.1509
0.3036
0.5000
0.6964
0.8491
0.9408
0.9824
0.9963
0.9995
1.0000
0.0743
0.2839
0.5614
0.7899
0.9209
0.9765
0.9944
0.9989
0.9998
1.0000
0.0541
0.2272
0.4868
0.7291
0.8866
0.9622
0.9899
0.9979
0.9996
1.0000
0.0281
0.1407
0.3518
0.5981
0.7982
0.9183
0.9733
0.9930
0.9985
0.9998
1.0000
0.0100
0.0635
0.1971
0.4050
0.6302
0.8103
0.9204
0.9729
0.9925
0.9984
0.9997
1.0000
0.0033
0.0261
0.0994
0.2459
0.4499
0.6598
0.8247
0.9256
0.9743
0.9929
0.9984
0.9997
1.0000
0.0015
0.0137
0.0594
0.1659
0.3391
0.5469
0.7374
0.8735
0.9500
0.9841
0.9960
0.9992
0.9999
1.0000
0.0010
0.0098
0.0451
0.1339
0.2892
0.4900
0.6881
0.8406
0.9329
0.9771
0.9938
0.9987
0.9998
1.0000
0.0003
0.0033
0.0183
0.0651
0.1666
0.3288
0.5272
0.7161
0.8577
0.9417
0.9809
0.9951
0.9991
0.9999
1.0000
0.0001
0.0010
0.0066
0.0281
0.0853
0.1976
0.3660
0.5629
0.7441
0.8759
0.9514
0.9851
0.9965
0.9994
0.9999
1.0000
0.0000
0.0003
0.0021
0.0106
0.0384
0.1051
0.2272
0.4018
0.5982
0.7728
0.8949
0.9616
0.9894
0.9979
0.9997
1.0000
0.0631
0.2525
0.5198
0.7556
0.9013
0.9681
0.9917
0.9983
0.9997
1.0000
0.0451
0.1983
0.4435
0.6887
0.8604
0.9496
0.9853
0.9965
0.9993
0.9999
1.0000
0.0225
0.1182
0.3096
0.5489
0.7582
0.8943
0.9623
0.9891
0.9974
0.9995
0.9999
1.0000
0.0075
0.0501
0.1637
0.3530
0.5739
0.7653
0.8929
0.9598
0.9876
0.9969
0.9994
0.9999
1.0000
0.0023
0.0193
0.0774
0.2019
0.3887
0.5968
0.7752
0.8954
0.9597
0.9873
0.9968
0.9993
0.9999
1.0000
0.0010
0.0096
0.0442
0.1304
0.2814
0.4777
0.6739
0.8281
0.9245
0.9727
0.9920
0.9981
0.9997
1.0000
0.0007
0.0067
0.0327
0.1028
0.2348
0.4197
0.6188
0.7872
0.9006
0.9617
0.9880
0.9970
0.9994
0.9999
1.0000
0.0002
0.0021
0.0123
0.0464
0.1260
0.2639
0.4478
0.6405
0.8011
0.9081
0.9652
0.9894
0.9975
0.9995
0.9999
1.0000
0.0000
0.0006
0.0041
0.0184
0.0596
0.1471
0.2902
0.4743
0.6626
0.8166
0.9174
0.9699
0.9914
0.9981
0.9997
1.0000
0.0000
0.0001
0.0012
0.0064
0.0245
0.0717
0.1662
0.3145
0.5000
0.6855
0.8338
0.9283
0.9755
0.9936
0.9988
0.9999
1.0000
0.550
0.0000
0.0001
0.0011
0.0063
0.0255
0.0769
0.1818
0.3465
0.5478
0.7392
0.8796
0.9576
0.9893
0.9983
0.9999
1.0000
0.0000
0.0001
0.0006
0.0035
0.0149
0.0486
0.1241
0.2559
0.4371
0.6340
0.8024
0.9147
0.9719
0.9934
0.9990
0.9999
1.0000
0.0000
0.0003
0.0019
0.0086
0.0301
0.0826
0.1834
0.3374
0.5257
0.7098
0.8529
0.9404
0.9816
0.9959
0.9994
1.0000
0.600 0.650 2/3
0.700 0.750 0.800
5/6
0.850 0.900 0.950
0.0000
0.0003
0.0019
0.0093
0.0338
0.0950
0.2131
0.3902
0.5968
0.7827
0.9095
0.9729
0.9948
0.9995
1.0000
0.0000
0.0001
0.0005
0.0028
0.0124
0.0422
0.1132
0.2452
0.4357
0.6481
0.8273
0.9383
0.9858
0.9984
1.0000
0.0000
0.0003
0.0018
0.0085
0.0308
0.0882
0.2030
0.3816
0.5959
0.7908
0.9206
0.9806
0.9977
1.0000
0.0000
0.0001
0.0007
0.0037
0.0152
0.0500
0.1311
0.2784
0.4845
0.7031
0.8732
0.9647
0.9953
1.0000
0.0000
0.0001
0.0008
0.0042
0.0173
0.0566
0.1484
0.3135
0.5387
0.7639
0.9198
0.9866
1.0000
0.0000
0.0001
0.0008
0.0042
0.0181
0.0611
0.1642
0.3518
0.6020
0.8329
0.9648
1.0000
0.0000
0.0002
0.0013
0.0066
0.0274
0.0898
0.2315
0.4678
0.7404
0.9351
1.0000
0.0000
0.0001
0.0006
0.0036
0.0168
0.0617
0.1773
0.3958
0.6814
0.9126
1.0000
0.0000
0.0003
0.0022
0.0127
0.0556
0.1841
0.4510
0.7941
1.0000
0.0000
0.0001
0.0006
0.0055
0.0362
0.1710
0.5367
1.0000
0.0000
0.0001
0.0009
0.0049
0.0191
0.0583
0.1423
0.2839
0.4728
0.6712
0.8334
0.9349
0.9817
0.9967
0.9997
1.0000
0.0000
0.0002
0.0013
0.0062
0.0229
0.0671
0.1594
0.3119
0.5100
0.7108
0.8661
0.9549
0.9902
0.9990
1.0000
0.0000
0.0001
0.0008
0.0040
0.0159
0.0500
0.1265
0.2626
0.4531
0.6609
0.8341
0.9406
0.9863
0.9985
1.0000
0.0000
0.0003
0.0016
0.0071
0.0257
0.0744
0.1753
0.3402
0.5501
0.7541
0.9006
0.9739
0.9967
1.0000
0.0000
0.0003
0.0016
0.0075
0.0271
0.0796
0.1897
0.3698
0.5950
0.8029
0.9365
0.9900
1.0000
0.0000
0.0002
0.0015
0.0070
0.0267
0.0817
0.2018
0.4019
0.6482
0.8593
0.9719
1.0000
0.0000
0.0004
0.0021
0.0101
0.0378
0.1134
0.2709
0.5132
0.7728
0.9459
1.0000
0.0000
0.0002
0.0011
0.0056
0.0235
0.0791
0.2101
0.4386
0.7161
0.9257
1.0000
0.0000
0.0001
0.0005
0.0033
0.0170
0.0684
0.2108
0.4853
0.8147
1.0000
0.0000
0.0001
0.0009
0.0070
0.0429
0.1892
0.5599
1.0000
0.0000
0.0001
0.0005
0.0025
0.0106
0.0348
0.0919
0.1989
0.3595
0.5522
0.7361
0.8740
0.9536
0.9877
0.9979
0.9998
1.0000
0.0000
0.0001
0.0006
0.0030
0.0120
0.0383
0.0994
0.2128
0.3812
0.5803
0.7652
0.8972
0.9673
0.9933
0.9993
1.0000
0.0000
0.0003
0.0019
0.0080
0.0273
0.0755
0.1719
0.3261
0.5223
0.7186
0.8696
0.9558
0.9904
0.9990
1.0000
0.0000
0.0001
0.0007
0.0032
0.0127
0.0403
0.1046
0.2248
0.4032
0.6113
0.7981
0.9226
0.9807
0.9977
1.0000
0.0000
0.0001
0.0006
0.0031
0.0124
0.0402
0.1071
0.2347
0.4261
0.6470
0.8363
0.9499
0.9925
1.0000
0.0000
0.0001
0.0005
0.0026
0.0109
0.0377
0.1057
0.2418
0.4511
0.6904
0.8818
0.9775
1.0000
0.0000
0.0001
0.0007
0.0035
0.0147
0.0504
0.1396
0.3113
0.5565
0.8017
0.9549
1.0000
0.0000
0.0003
0.0017
0.0083
0.0319
0.0987
0.2444
0.4802
0.7475
0.9369
1.0000
0.0000
0.0001
0.0008
0.0047
0.0221
0.0826
0.2382
0.5182
0.8332
1.0000
0.0000
0.0001
0.0012
0.0088
0.0503
0.2078
0.5819
1.0000
CUMULATIVE BINOMIAL PROBABILITY
15
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
n
18
16
0.050 0.100
0.150 1/6
0.200
0.250 0.300
1/3
0.350 0.400 0.450
0.500 0.550
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
0.3972
0.7735
0.9419
0.9891
0.9985
0.9998
1.0000
0.0536
0.2241
0.4794
0.7202
0.8794
0.9581
0.9882
0.9973
0.9995
0.9999
1.0000
0.0180
0.0991
0.2713
0.5010
0.7164
0.8671
0.9487
0.9837
0.9957
0.9991
0.9998
1.0000
0.0056
0.0395
0.1353
0.3057
0.5187
0.7175
0.8610
0.9431
0.9807
0.9946
0.9988
0.9998
1.0000
0.0016
0.0142
0.0600
0.1646
0.3327
0.5344
0.7217
0.8593
0.9404
0.9790
0.9939
0.9986
0.9997
1.0000
0.0007
0.0068
0.0326
0.1017
0.2311
0.4122
0.6085
0.7767
0.8924
0.9567
0.9856
0.9961
0.9991
0.9999
1.0000
0.0004
0.0046
0.0236
0.0783
0.1886
0.3550
0.5491
0.7283
0.8609
0.9403
0.9788
0.9938
0.9986
0.9997
1.0000
0.0001
0.0013
0.0082
0.0328
0.0942
0.2088
0.3743
0.5634
0.7368
0.8653
0.9424
0.9797
0.9942
0.9987
0.9998
1.0000
0.0000
0.0003
0.0025
0.0120
0.0411
0.1077
0.2258
0.3915
0.5778
0.7473
0.8720
0.9463
0.9817
0.9951
0.9990
0.9999
1.0000
0.0000
0.0001
0.0007
0.0038
0.0154
0.0481
0.1189
0.2403
0.4073
0.5927
0.7597
0.8811
0.9519
0.9846
0.9962
0.9993
0.9999
1.0000
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
0.3774
0.7547
0.9335
0.9868
0.9980
0.9998
1.0000
0.0144
0.0829
0.2369
0.4551
0.6733
0.8369
0.9324
0.9767
0.9933
0.9984
0.9997
1.0000
0.0042
0.0310
0.1113
0.2631
0.4654
0.6678
0.8251
0.9225
0.9713
0.9911
0.9977
0.9995
0.9999
1.0000
0.0011
0.0104
0.0462
0.1332
0.2822
0.4739
0.6655
0.8180
0.9161
0.9674
0.9895
0.9972
0.9994
0.9999
1.0000
0.0005
0.0047
0.0240
0.0787
0.1879
0.3519
0.5431
0.7207
0.8538
0.9352
0.9759
0.9926
0.9981
0.9996
0.9999
1.0000
0.0003
0.0031
0.0170
0.0591
0.1500
0.2968
0.4812
0.6656
0.8145
0.9125
0.9653
0.9886
0.9969
0.9993
0.9999
1.0000
0.0001
0.0008
0.0055
0.0230
0.0696
0.1629
0.3081
0.4878
0.6675
0.8139
0.9115
0.9648
0.9884
0.9969
0.9994
0.9999
1.0000
0.0000
0.0002
0.0015
0.0077
0.0280
0.0777
0.1727
0.3169
0.4940
0.6710
0.8159
0.9129
0.9658
0.9891
0.9972
0.9995
0.9999
1.0000
0.1501
0.4503
0.7338
0.9018
0.9718
0.9936
0.9988
0.9998
1.0000
0.1351
0.4203
0.7054
0.8850
0.9648
0.9914
0.9983
0.9997
1.0000
0.0456
0.1985
0.4413
0.6841
0.8556
0.9463
0.9837
0.9959
0.9992
0.9999
1.0000
0.0376
0.1728
0.4027
0.6479
0.8318
0.9347
0.9794
0.9947
0.9989
0.9998
1.0000
0.0313
0.1502
0.3643
0.6070
0.8011
0.9176
0.9719
0.9921
0.9982
0.9996
0.9999
1.0000
0.0000
0.0004
0.0022
0.0096
0.0318
0.0835
0.1796
0.3238
0.5000
0.6762
0.8204
0.9165
0.9682
0.9904
0.9978
0.9996
1.0000
0.0000
0.0001
0.0010
0.0049
0.0183
0.0537
0.1280
0.2527
0.4222
0.6085
0.7742
0.8923
0.9589
0.9880
0.9975
0.9997
1.0000
0.0000
0.0001
0.0005
0.0028
0.0109
0.0342
0.0871
0.1841
0.3290
0.5060
0.6831
0.8273
0.9223
0.9720
0.9923
0.9985
0.9998
1.0000
0.600 0.650 2/3
0.700 0.750 0.800
5/6
0.850 0.900 0.950
0.0000
0.0002
0.0013
0.0058
0.0203
0.0576
0.1347
0.2632
0.4366
0.6257
0.7912
0.9058
0.9672
0.9918
0.9987
0.9999
1.0000
0.0000
0.0003
0.0014
0.0062
0.0212
0.0597
0.1391
0.2717
0.4509
0.6450
0.8114
0.9217
0.9764
0.9954
0.9996
1.0000
0.0000
0.0001
0.0009
0.0039
0.0144
0.0433
0.1076
0.2233
0.3915
0.5878
0.7689
0.8983
0.9674
0.9932
0.9993
1.0000
0.0000
0.0003
0.0014
0.0061
0.0210
0.0596
0.1407
0.2783
0.4656
0.6673
0.8354
0.9400
0.9858
0.9984
1.0000
0.0000
0.0002
0.0012
0.0054
0.0193
0.0569
0.1390
0.2825
0.4813
0.6943
0.8647
0.9605
0.9944
1.0000
0.0000
0.0002
0.0009
0.0043
0.0163
0.0513
0.1329
0.2836
0.4990
0.7287
0.9009
0.9820
1.0000
0.0000
0.0002
0.0011
0.0053
0.0206
0.0653
0.1682
0.3521
0.5973
0.8272
0.9624
1.0000
0.0000
0.0001
0.0005
0.0027
0.0118
0.0419
0.1206
0.2798
0.5203
0.7759
0.9464
1.0000
0.0000
0.0002
0.0012
0.0064
0.0282
0.0982
0.2662
0.5497
0.8499
1.0000
0.0000
0.0002
0.0015
0.0109
0.0581
0.2265
0.6028
1.0000
0.0000
0.0001
0.0006
0.0031
0.0116
0.0352
0.0885
0.1861
0.3325
0.5122
0.6919
0.8371
0.9304
0.9770
0.9945
0.9992
0.9999
1.0000
0.0000
0.0001
0.0007
0.0031
0.0114
0.0347
0.0875
0.1855
0.3344
0.5188
0.7032
0.8500
0.9409
0.9830
0.9969
0.9997
1.0000
0.0000
0.0001
0.0004
0.0019
0.0074
0.0241
0.0648
0.1462
0.2793
0.4569
0.6481
0.8121
0.9213
0.9760
0.9953
0.9995
1.0000
0.0000
0.0001
0.0006
0.0028
0.0105
0.0326
0.0839
0.1820
0.3345
0.5261
0.7178
0.8668
0.9538
0.9896
0.9989
1.0000
0.0000
0.0001
0.0005
0.0023
0.0089
0.0287
0.0775
0.1749
0.3322
0.5346
0.7369
0.8887
0.9690
0.9958
1.0000
0.0000
0.0003
0.0016
0.0067
0.0233
0.0676
0.1631
0.3267
0.5449
0.7631
0.9171
0.9856
1.0000
0.0000
0.0001
0.0004
0.0018
0.0079
0.0281
0.0824
0.1989
0.3930
0.6357
0.8498
0.9687
1.0000
0.0000
0.0001
0.0008
0.0041
0.0163
0.0537
0.1444
0.3159
0.5587
0.8015
0.9544
1.0000
0.0000
0.0003
0.0017
0.0086
0.0352
0.1150
0.2946
0.5797
0.8649
1.0000
0.0002
0.0020
0.0132
0.0665
0.2453
0.6226
1.0000
CUMULATIVE BINOMIAL PROBABILITY
19
p
x
n
20
p
x
0.150 1/6
0.200
0.250 0.300
1/3
0.350 0.400 0.450
0.3585
0.7358
0.9245
0.9841
0.9974
0.9997
1.0000
0.0388
0.1756
0.4049
0.6477
0.8298
0.9327
0.9781
0.9941
0.9987
0.9998
1.0000
0.0115
0.0692
0.2061
0.4114
0.6296
0.8042
0.9133
0.9679
0.9900
0.9974
0.9994
0.9999
1.0000
0.0032
0.0243
0.0913
0.2252
0.4148
0.6172
0.7858
0.8982
0.9591
0.9861
0.9961
0.9991
0.9998
1.0000
0.0003
0.0033
0.0176
0.0604
0.1515
0.2972
0.4793
0.6615
0.8095
0.9081
0.9624
0.9870
0.9963
0.9991
0.9998
1.0000
0.0002
0.0021
0.0121
0.0444
0.1182
0.2454
0.4166
0.6010
0.7624
0.8782
0.9468
0.9804
0.9940
0.9985
0.9998
1.0000
0.1216
0.3917
0.6769
0.8670
0.9568
0.9887
0.9976
0.9996
0.9999
1.0000
0.0261
0.1304
0.3287
0.5665
0.7687
0.8982
0.9629
0.9887
0.9972
0.9994
0.9999
1.0000
0.0008
0.0076
0.0355
0.1071
0.2375
0.4164
0.6080
0.7723
0.8867
0.9520
0.9829
0.9949
0.9987
0.9997
1.0000
0.0000
0.0005
0.0036
0.0160
0.0510
0.1256
0.2500
0.4159
0.5956
0.7553
0.8725
0.9435
0.9790
0.9935
0.9984
0.9997
1.0000
0.0000
0.0001
0.0009
0.0049
0.0189
0.0553
0.1299
0.2520
0.4143
0.5914
0.7507
0.8692
0.9420
0.9786
0.9936
0.9985
0.9997
1.0000
0.500 0.550
0.0000
0.0002
0.0013
0.0059
0.0207
0.0577
0.1316
0.2517
0.4119
0.5881
0.7483
0.8684
0.9423
0.9793
0.9941
0.9987
0.9998
1.0000
0.0000
0.0003
0.0015
0.0064
0.0214
0.0580
0.1308
0.2493
0.4086
0.5857
0.7480
0.8701
0.9447
0.9811
0.9951
0.9991
0.9999
1.0000
0.600 0.650 2/3
0.0000
0.0003
0.0016
0.0065
0.0210
0.0565
0.1275
0.2447
0.4044
0.5841
0.7500
0.8744
0.9490
0.9840
0.9964
0.9995
1.0000
0.0000
0.0003
0.0015
0.0060
0.0196
0.0532
0.1218
0.2376
0.3990
0.5834
0.7546
0.8818
0.9556
0.9879
0.9979
0.9998
1.0000
0.0000
0.0002
0.0009
0.0037
0.0130
0.0376
0.0919
0.1905
0.3385
0.5207
0.7028
0.8485
0.9396
0.9824
0.9967
0.9997
1.0000
0.700 0.750 0.800
0.0000
0.0003
0.0013
0.0051
0.0171
0.0480
0.1133
0.2277
0.3920
0.5836
0.7625
0.8929
0.9645
0.9924
0.9992
1.0000
0.0000
0.0002
0.0009
0.0039
0.0139
0.0409
0.1018
0.2142
0.3828
0.5852
0.7748
0.9087
0.9757
0.9968
1.0000
0.0000
0.0001
0.0006
0.0026
0.0100
0.0321
0.0867
0.1958
0.3704
0.5886
0.7939
0.9308
0.9885
1.0000
5/6
0.0000
0.0001
0.0006
0.0028
0.0113
0.0371
0.1018
0.2313
0.4335
0.6713
0.8696
0.9739
1.0000
0.850 0.900 0.950
0.0000
0.0002
0.0013
0.0059
0.0219
0.0673
0.1702
0.3523
0.5951
0.8244
0.9612
1.0000
0.0000
0.0001
0.0004
0.0024
0.0113
0.0432
0.1330
0.3231
0.6083
0.8784
1.0000
0.0000
0.0003
0.0026
0.0159
0.0755
0.2642
0.6415
1.0000
17
CUMULATIVE BINOMIAL PROBABILITY
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.050 0.100
The Poisson distribution: cumulative probabilities
x
λr
P (X x) = ∑ e–λ ––
r!
r=0
xλ
0.01
0
1
2
3
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.9139
0.9962
0.9999
1.0000
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
0.9048
0.9953
0.9998
1.0000
.....
.....
.....
.....
0.8187
0.9825
0.9989
0.9999
1.0000
.....
.....
.....
0.7408
0.9631
0.9964
0.9997
1.0000
.....
.....
.....
0.6703
0.9384
0.9921
0.9992
0.9999
1.0000
.....
.....
0.6065
0.9098
0.9856
0.9982
0.9998
1.0000
.....
.....
0.5488
0.8781
0.9769
0.9966
0.9996
1.0000
.....
.....
0.4966
0.8442
0.9659
0.9942
0.9992
0.9999
1.0000
.....
0.4493
0.8088
0.9526
0.9909
0.9986
0.9998
1.0000
.....
0.4066
0.7725
0.9371
0.9865
0.9977
0.9997
1.0000
.....
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
0.3329
0.6990
0.9004
0.9743
0.9946
0.9990
0.9999
1.0000
.....
.....
0.3012
0.6626
0.8795
0.9662
0.9923
0.9985
0.9997
1.0000
.....
.....
0.2725
0.6268
0.8571
0.9569
0.9893
0.9978
0.9996
0.9999
1.0000
.....
0.2466
0.5918
0.8335
0.9463
0.9857
0.9968
0.9994
0.9999
1.0000
.....
0.2231
0.5578
0.8088
0.9344
0.9814
0.9955
0.9991
0.9998
1.0000
.....
0.2019
0.5249
0.7834
0.9212
0.9763
0.9940
0.9987
0.9997
1.0000
.....
0.1827
0.4932
0.7572
0.9068
0.9704
0.9920
0.9981
0.9996
0.9999
1.0000
0.1653
0.4628
0.7306
0.8913
0.9636
0.9896
0.9974
0.9994
0.9999
1.0000
0.1496
0.4337
0.7037
0.8747
0.9559
0.9868
0.9966
0.9992
0.9998
1.0000
x λ 2.00
2.10
2.20
2.30
2.40
2.50
2.60
2.70
2.80
2.90
0
1
2
3
4
5
6
7
8
9
10
11
12
0.1225
0.3796
0.6496
0.8386
0.9379
0.9796
0.9941
0.9985
0.9997
0.9999
1.0000
.....
.....
0.1108
0.3546
0.6227
0.8194
0.9275
0.9751
0.9925
0.9980
0.9995
0.9999
1.0000
.....
.....
0.1003
0.3309
0.5960
0.7993
0.9162
0.9700
0.9906
0.9974
0.9994
0.9999
1.0000
.....
.....
0.0907
0.3084
0.5697
0.7787
0.9041
0.9643
0.9884
0.9967
0.9991
0.9998
1.0000
.....
.....
0.0821
0.2873
0.5438
0.7576
0.8912
0.9580
0.9858
0.9958
0.9989
0.9997
0.9999
1.0000
.....
0.0743
0.2674
0.5184
0.7360
0.8774
0.9510
0.9828
0.9947
0.9985
0.9996
0.9999
1.0000
.....
0.0672
0.2487
0.4936
0.7141
0.8629
0.9433
0.9794
0.9934
0.9981
0.9995
0.9999
1.0000
.....
0.0608
0.2311
0.4695
0.6919
0.8477
0.9349
0.9756
0.9919
0.9976
0.9993
0.9998
1.0000
.....
0.0550
0.2146
0.4460
0.6696
0.8318
0.9258
0.9713
0.9901
0.9969
0.9991
0.9998
0.9999
1.0000
xλ
0
1
2
3
4
5
6
7
x λ 1.00
18
0
1
2
3
4
5
6
7
8
9
0.3679
0.7358
0.9197
0.9810
0.9963
0.9994
0.9999
1.0000
.....
.....
0.1353
0.4060
0.6767
0.8571
0.9473
0.9834
0.9955
0.9989
0.9998
1.0000
.....
.....
.....
3.10
3.20
3.30
3.40
3.50
3.60
3.70
3.80
3.90
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.0450
0.1847
0.4012
0.6248
0.7982
0.9057
0.9612
0.9858
0.9953
0.9986
0.9996
0.9999
1.0000
.....
.....
0.0408
0.1712
0.3799
0.6025
0.7806
0.8946
0.9554
0.9832
0.9943
0.9982
0.9995
0.9999
1.0000
.....
.....
0.0369
0.1586
0.3594
0.5803
0.7626
0.8829
0.9490
0.9802
0.9931
0.9978
0.9994
0.9998
1.0000
.....
.....
0.0334
0.1468
0.3397
0.5584
0.7442
0.8705
0.9421
0.9769
0.9917
0.9973
0.9992
0.9998
0.9999
1.0000
.....
0.0302
0.1359
0.3208
0.5366
0.7254
0.8576
0.9347
0.9733
0.9901
0.9967
0.9990
0.9997
0.9999
1.0000
.....
0.0273
0.1257
0.3027
0.5152
0.7064
0.8441
0.9267
0.9692
0.9883
0.9960
0.9987
0.9996
0.9999
1.0000
.....
0.0247
0.1162
0.2854
0.4942
0.6872
0.8301
0.9182
0.9648
0.9863
0.9952
0.9984
0.9995
0.9999
1.0000
.....
0.0224
0.1074
0.2689
0.4735
0.6678
0.8156
0.9091
0.9599
0.9840
0.9942
0.9981
0.9994
0.9998
1.0000
.....
0.0202
0.0992
0.2531
0.4532
0.6484
0.8006
0.8995
0.9546
0.9815
0.9931
0.9977
0.9993
0.9998
0.9999
1.0000
x λ 4.00
4.10
4.20
4.30
4.40
4.50
4.60
4.70
4.80
4.90
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.0166
0.0845
0.2238
0.4142
0.6093
0.7693
0.8786
0.9427
0.9755
0.9905
0.9966
0.9989
0.9997
0.9999
1.0000
.....
.....
0.0150
0.0780
0.2102
0.3954
0.5898
0.7531
0.8675
0.9361
0.9721
0.9889
0.9959
0.9986
0.9996
0.9999
1.0000
.....
.....
0.0136
0.0719
0.1974
0.3772
0.5704
0.7367
0.8558
0.9290
0.9683
0.9871
0.9952
0.9983
0.9995
0.9998
1.0000
.....
.....
0.0123
0.0663
0.1851
0.3594
0.5512
0.7199
0.8436
0.9214
0.9642
0.9851
0.9943
0.9980
0.9993
0.9998
0.9999
1.0000
.....
0.0111
0.0611
0.1736
0.3423
0.5321
0.7029
0.8311
0.9134
0.9597
0.9829
0.9933
0.9976
0.9992
0.9997
0.9999
1.0000
.....
0.0101
0.0563
0.1626
0.3257
0.5132
0.6858
0.8180
0.9049
0.9549
0.9805
0.9922
0.9971
0.9990
0.9997
0.9999
1.0000
.....
0.0091
0.0518
0.1523
0.3097
0.4946
0.6684
0.8046
0.8960
0.9497
0.9778
0.9910
0.9966
0.9988
0.9996
0.9999
1.0000
.....
0.0082
0.0477
0.1425
0.2942
0.4763
0.6510
0.7908
0.8867
0.9442
0.9749
0.9896
0.9960
0.9986
0.9995
0.9999
1.0000
.....
0.0074
0.0439
0.1333
0.2793
0.4582
0.6335
0.7767
0.8769
0.9382
0.9717
0.9880
0.9953
0.9983
0.9994
0.9998
0.9999
1.0000
0.0498
0.1991
0.4232
0.6472
0.8153
0.9161
0.9665
0.9881
0.9962
0.9989
0.9997
0.9999
1.0000
.....
.....
0.0183
0.0916
0.2381
0.4335
0.6288
0.7851
0.8893
0.9489
0.9786
0.9919
0.9972
0.9991
0.9997
0.9999
1.0000
.....
.....
CUMULATIVE POISSON PROBABILITY
0.9900 0.9802 0.9704 0.9608 0.9512 0.9418 0.9324 0.9231
1.0000 0.9998 0.9996 0.9992 0.9988 0.9983 0.9977 0.9970
. . . . . 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0000 1.0000
x λ 3.00
5.10
5.20
5.30
5.40
5.50
5.60
5.70
5.80
5.90
x λ 7.00
7.10
7.20
7.30
7.40
7.50
7.60
7.70
7.80
7.90
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
0.0067
0.0404
0.1247
0.2650
0.4405
0.6160
0.7622
0.8666
0.9319
0.9682
0.9863
0.9945
0.9980
0.9993
0.9998
0.9999
1.0000
.....
0.0061
0.0372
0.1165
0.2513
0.4231
0.5984
0.7474
0.8560
0.9252
0.9644
0.9844
0.9937
0.9976
0.9992
0.9997
0.9999
1.0000
.....
0.0055
0.0342
0.1088
0.2381
0.4061
0.5809
0.7324
0.8449
0.9181
0.9603
0.9823
0.9927
0.9972
0.9990
0.9997
0.9999
1.0000
.....
0.0050
0.0314
0.1016
0.2254
0.3895
0.5635
0.7171
0.8335
0.9106
0.9559
0.9800
0.9916
0.9967
0.9988
0.9996
0.9999
1.0000
.....
0.0045
0.0289
0.0948
0.2133
0.3733
0.5461
0.7017
0.8217
0.9027
0.9512
0.9775
0.9904
0.9962
0.9986
0.9995
0.9998
0.9999
1.0000
0.0041
0.0266
0.0884
0.2017
0.3575
0.5289
0.6860
0.8095
0.8944
0.9462
0.9747
0.9890
0.9955
0.9983
0.9994
0.9998
0.9999
1.0000
0.0037
0.0244
0.0824
0.1906
0.3422
0.5119
0.6703
0.7970
0.8857
0.9409
0.9718
0.9875
0.9949
0.9980
0.9993
0.9998
0.9999
1.0000
0.0033
0.0224
0.0768
0.1800
0.3272
0.4950
0.6544
0.7841
0.8766
0.9352
0.9686
0.9859
0.9941
0.9977
0.9991
.09997
0.9999
1.0000
0.0030
0.0206
0.0715
0.1700
0.3127
0.4783
0.6384
0.7710
0.8672
0.9292
0.9651
0.9841
0.9932
0.9973
0.9990
0.9996
0.9999
1.0000
0.0027
0.0189
0.0666
0.1604
0.2987
0.4619
0.6224
0.7576
0.8574
0.9228
0.9614
0.9821
0.9922
0.9969
0.9988
0.9996
0.9999
1.0000
xλ
6.00
6.10
6.20
6.30
6.40
6.50
6.60
6.70
6.80
6.90
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0.0009
0.0073
0.0296
0.0818
0.1730
0.3007
0.4497
0.5987
0.7291
0.8305
0.9015
0.9467
0.9730
0.9872
0.9943
0.9976
0.9990
0.9996
0.9999
1.0000
.....
.....
0.0008
0.0067
0.0275
0.0767
0.1641
0.2881
0.4349
0.5838
0.7160
0.8202
0.8942
0.9420
0.9703
0.9857
0.9935
0.9972
0.9989
0.9996
0.9998
0.9999
1.0000
.....
0.0007
0.0061
0.0255
0.0719
0.1555
0.2759
0.4204
0.5689
0.7027
0.8096
0.8867
0.9371
0.9673
0.9841
0.9927
0.9969
0.9987
0.9995
0.9998
0.9999
1.0000
.....
0.0007
0.0056
0.0236
0.0674
0.1473
0.2640
0.4060
0.5541
0.6892
0.7988
0.8788
0.9319
0.9642
0.9824
0.9918
0.9964
0.9985
0.9994
0.9998
0.9999
1.0000
.....
00006
0.0051
0.0219
0.0632
0.1395
0.2526
0.3920
0.5393
0.6757
0.7877
0.8707
0.9265
0.9609
0.9805
0.9908
0.9959
0.9983
0.9993
0.9997
0.9999
1.0000
.....
0.0006
0.0047
0.0203
0.0591
0.1321
0.2414
0.3782
0.5246
0.6620
0.7764
0.8622
0.9208
0.9573
0.9784
0.9897
0.9954
0.9980
0.9992
0.9997
0.9999
1.0000
.....
0.0005
0.0043
0.0188
0.0554
0.1249
0.2307
0.3646
0.5100
0.6482
0.7649
0.8535
0.9148
0.9536
0.9762
0.9886
0.9948
09978
0.9991
0.9996
0.9999
1.0000
.....
0.0005
0.0039
0.0174
0.0518
0.1181
0.2203
0.3514
0.4956
0.6343
0.7531
0.8445
0.9085
0.9496
0.9739
0.9873
0.9941
0.9974
0.9989
0.9996
0.9998
0.9999
1.0000
0.0004
0.0036
0.0161
0.0485
0.1117
0.2103
0.3384
0.4812
0.6204
0.7411
0.8352
0.9020
0.9454
0.9714
0.9859
0.9934
0.9971
0.9988
0.9995
0.9998
0.9999
1.0000
0.0004
0.0033
0.0149
0.0453
0.1055
0.2006
0.3257
0.4670
0.6065
0.7290
0.8257
0.8952
0.9409
0.9687
0.9844
0.9926
09967
0.9986
0.9994
0.9998
0.9999
1.0000
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
0.0025
0.0174
0.0620
0.1512
0.2851
0.4457
0.6063
0.7440
0.8472
0.9161
0.9574
0.9799
0.9912
0.9964
0.9986
0.9995
0.9998
0.9999
1..0000
.....
0.0022
0.0159
0.0577
0.1425
0.2719
0.4298
0.5902
0.7301
0.8367
0.9090
0.9531
0.9776
0.9900
0.9958
0.9984
0.9994
0.9998
0.9999
1.0000
.....
0.0020
0.0146
0.0536
0.1342
0.2592
0.4141
0.5742
0.7160
0.8259
0.9016
0.9486
0.9750
0.9887
0.9952
0.9981
0.9993
0.9997
0.9999
1.0000
.....
0.0018
0.0134
0.0498
0.1264
0.2469
0.3988
0.5582
0.7017
0.8148
0.8939
0.9437
0.9723
0.9873
0.9945
0.9978
0.9992
0.9997
0.9999
1.0000
.....
0.0017
0.0123
0.0463
0.1189
0.2351
0.3837
0.5423
0.6873
0.8033
0.8858
0.9386
0.9693
0.9857
0.9937
0.9974
0.9990
0.9996
0.9999
1.0000
.....
0.0015
0.0113
0.0430
0.1118
0.2237
0.3690
0.5265
0.6728
0.7916
0.8774
0.9332
0.9661
0.9840
0.9929
0.9970
0.9988
0.9996
0.9998
0.9999
1.0000
0.0014
0.0103
0.0400
0.1052
0.2127
0.3547
0.5108
0.6581
0.7796
0.8686
0.9274
0.9627
0.9821
0.9920
0.9966
0.9986
0.9995
0.9998
0.9999
1.0000
0.0012
0.0095
0.0371
0.0988
0.2022
0.3406
0.4953
0.6433
0.7673
0.8596
0.9214
0.9591
0.9801
0.9909
0.9961
0.9984
0.9994
0.9998
0.9999
1.0000
0.0011
0.0087
0.0344
0.0928
0.1920
0.3270
0.4799
0.6285
0.7548
0.8502
0.9151
0.9552
0.9779
0.9898
0.9956
0.9982
0.9993
0.9997
0.9999
1.0000
0.0010
0.0080
0.0320
0.0871
0.1823
0.3137
0.4647
0.6136
0.7420
0.8405
0.9084
0.9510
0.9755
0.9885
0.9950
0.9979
0.9992
0.9997
0.9999
1.0000
x λ
8.00
8.10
8.20
8.30
8.40
8.50
8.60
8.70
8.80
8.90
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
0.0003
0.0030
0.0138
0.0424
0.0996
0.1912
0.3134
0.4530
0.5925
0.7166
0.8159
0.8881
0.9362
0.9658
0.9827
0.9918
0.9963
0.9984
0.9993
0.9997
0.9999
1.0000
.....
.....
0.0003
0.0028
0.0127
0.0396
0.0940
0.1822
0.3013
0.4391
0.5786
0.7041
0.8058
0.8807
0.9313
0.9628
0.9810
0.9908
0.9958
0.9982
0.9992
0.9997
0.9999
1.0000
.....
.....
0.0003
0.0025
0.0118
0.0370
0.0887
0.1736
0.2896
0.4254
0.5647
0.6915
0.7955
0.8731
0.9261
0.9595
0.9791
0.9898
0.9953
0.9979
0.9991
0.9997
0.9999
1.0000
.....
.....
0.0002
0.0023
0.0109
0.0346
0.0837
0.1653
0.2781
0.4119
0.5507
0.6788
0.7850
0.8652
0.9207
0.9561
0.9771
0.9887
0.9947
0.9977
0.9990
0.9996
0.9998
0.9999
1.0000
.....
0.0002
0.0021
0.0100
0.0323
0.0789
0.1573
0.2670
0.3987
0.5369
0.6659
0.7743
0.8571
0.9150
0.9524
0.9749
0.9875
0.9941
0.9973
0.9989
0.9995
0.9998
0.9999
1.0000
.....
0.0002
0.0019
0.0093
0.0301
0.0744
0.1496
0.2562
0.3856
0.5231
0.6530
0.7634
0.8487
0.9091
0.9486
0.9726
0.9862
0.9934
0.9970
0.9987
0.9995
0.9998
0.9999
1.0000
.....
0.0002
0.0018
0.0086
0.0281
0.0701
0.1422
0.2457
0.3728
0.5094
0.6400
0.7522
0.8400
0.9029
0.9445
0.9701
0.9848
0.9926
0.9966
0.9985
0.9994
0.9998
0.9999
1.0000
.....
0.0002
0.0016
0.0079
0.0262
0.0660
0.1352
0.2355
0.3602
0.4958
0.6269
0.7409
0.8311
0.8965
0.9403
0.9675
0.9832
0.9918
0.9962
0.9983
0.9993
0.9997
0.9999
1.0000
.....
0.0002
0.0015
0.0073
0.0244
0.0621
0.1284
0.2256
0.3478
0.4823
0.6137
0.7294
0.8220
0.8898
0.9358
0.9647
0.9816
0.9909
0.9957
0.9981
0.9992
0.9997
0.9999
1.0000
.....
0.0001
0.0014
0.0068
0.0228
0.0584
0.1219
0.2160
0.3357
0.4689
0.6006
0.7178
0.8126
0.8829
0.9311
0.9617
0.9798
0.9899
0.9952
0.9978
0.9991
0.9996
0.9998
0.9999
1.0000
CUMULATIVE POISSON PROBABILITY
19
x λ 5.00
9.00
9.10
9.20
9.30
9.40
9.50
9.60
9.70
9.80
9.90
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0.0001
0.0012
0.0062
0.0212
0.0550
0.1157
0.2068
0.3239
0.4557
0.5874
0.7060
0.8030
0.8758
0.9261
0.9585
0.9780
0.9889
0.9947
0.9976
0.9989
0.9996
0.9998
0.9999
1.0000
.....
0.0001
0.0011
0.0058
0.0198
0.0517
0.1098
0.1978
0.3123
0.4426
0.5742
0.6941
0.7932
0.8684
0.9210
0.9552
0.9760
0.9878
0.9941
0.9973
0.9988
0.9995
0.9998
0.9999
1.0000
.....
0.0001
0.0010
0.0053
0.0184
0.0486
0.1041
0.1892
0.3010
0.4296
0.5611
0.6820
0.7832
0.8607
0.9156
0.9517
0.9738
0.9865
0.9934
0.9969
0.9986
0.9994
0.9998
0.9999
1.0000
.....
0.0001
0.0009
0.0049
0.0172
0.0456
0.0986
0.1808
0.2900
0.4168
0.5479
0.6699
0.7730
0.8529
0.9100
0.9480
0.9715
0.9852
0.9927
0.9966
0.9985
0.9993
0.9997
0.9999
1.0000
.....
0.0001
0.0009
0.0045
0.0160
0.0429
0.0935
0.1727
0.2792
0.4042
0.5349
0.6576
0.7626
0.8448
0.9042
0.9441
0.9691
0.9838
0.9919
0.9962
0.9983
0.9992
0.9997
0.9999
1.0000
.....
0.0001
0.0008
0.0042
0.0149
0.0403
0.0885
0.1649
0.2687
0.3918
0.5218
0.6453
0.7520
0.8364
0.8981
0.9400
0.9665
0.9823
0.9911
0.9957
0.9980
0.9991
0.9996
0.9999
0.9999
1.0000
0.0001
0.0007
0.0038
0.0138
0.0378
0.0838
0.1574
0.2584
0.3796
0.5089
0.6329
0.7412
0.8279
0.8919
0.9357
0.9638
0.9806
0.9902
0.9952
0.9978
0.9990
0.9996
0.9998
0.9999
1.0000
0.0001
0.0007
0.0035
0.0129
0.0355
0.0793
0.1502
0.2485
0.3676
0.4960
0.6205
0.7303
0.8191
0.8853
0.9312
0.9609
0.9789
0.9892
0.9947
0.9975
0.9989
0.9995
0.9998
0.9999
1.0000
0.0001
0.0006
0.0033
0.0120
0.0333
0.0750
0.1433
0.2388
0.3558
0.4832
0.6080
0.7193
0.8101
0.8786
0.9265
0.9579
0.9770
0.9881
0.9941
0.9972
0.9987
0.9995
0.9998
0.9999
1.0000
0.0001
0.0005
0.0030
0.0111
0.0312
0.0710
0.1366
0.2294
0.3442
0.4705
0.5955
0.7081
0.8009
0.8716
0.9216
0.9546
0.9751
0.9870
0.9935
0.9969
0.9986
0.9994
0.9997
0.9999
1.0000
x λ
10.00
10.10
10.20
10.30
10.40
10.50
10.60
10.70
10.80
10.90
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
0.0000
0.0005
0.0028
0.0103
0.0293
0.0671
0.1301
0.2202
0.3328
0.4579
0.5830
0.6968
0.7916
0.8645
0.9165
0.9513
0.9730
0.9857
0.9928
0.9965
0.9984
0.9993
0.9997
0.9999
1.0000
.....
.....
0.0000
0.0005
0.0026
0.0096
0.0274
0.0634
0.1240
0.2113
0.3217
0.4455
0.5705
0.6853
0.7820
0.8571
0.9112
0.9477
0.9707
0.9844
0.9921
0.9962
0.9982
0.9992
0.9997
0.9999
0.9999
1.0000
.....
0.0000
0.0004
0.0023
0.0089
0.0257
0.0599
0.1180
0.2027
0.3108
0.4332
0.5580
0.6738
0.7722
0.8494
0.9057
0.9440
0.9684
0.9830
0.9913
0.9957
0.9980
0.9991
0.9996
0.9998
0.9999
1.0000
.....
0.0000
0.0004
0.0022
0.0083
0.0241
0.0566
0.1123
0.1944
0.3001
0.4210
0.5456
0.6622
0.7623
0.8416
0.9000
0.9400
0.9658
0.9815
0.9904
0.9953
0.9978
0.9990
0.9996
0.9998
0.9999
1.0000
.....
0.0000
0.0003
0.0020
0.0077
0.0225
0.0534
0.1069
0.1863
0.2896
0.4090
0.5331
0.6505
0.7522
0.8336
0.8940
0.9359
0.9632
0.9799
0.9895
0.9948
0.9975
0.9989
0.9995
0.9998
0.9999
1.0000
.....
0.0000
0.0003
0.0018
0.0071
0.0211
0.0504
0.1016
0.1785
0.2794
0.3971
0.5207
0.6387
0.7420
0.8253
0.8879
0.9317
0.9604
0.9781
0.9885
0.9942
0.9972
0.9987
0.9994
0.9998
0.9999
1.0000
.....
0.0000
0.0003
0.0017
0.0066
0.0197
0.0475
0.0966
0.1710
0.2694
0.3854
0.5084
0.6269
0.7316
0.8169
0.8815
0.9272
0.9574
0.9763
0.9874
0.9936
0.9969
0.9986
0.9994
0.9997
0.9999
1.0000
.....
0.0000
0.0003
0.0016
0.0062
0.0185
0.0448
0.0918
0.1636
0.2597
0.3739
0.4961
0.6150
0.7210
0.8083
0.8750
0.9225
0.9543
0.9744
0.9863
0.9930
0.9966
0.9984
0.9993
0.9997
0.9999
0.9999
1.0000
0.0000
0.0002
0.0014
0.0057
0.0173
0.0423
0.0872
0.1566
0.2502
0.3626
0.4840
0.6031
0.7104
0.7995
0.8682
0.9177
0.9511
0.9723
0.9850
0.9923
0.9962
0.9982
0.9992
0.9996
0.9998
0.9999
1.0000
0.0000
0.0002
0.0013
0.0053
0.0162
0.0398
0.0828
0.1498
0.2410
0.3515
0.4719
0.5912
0.6996
0.7905
0.8612
0.9126
0.9477
0.9701
0.9837
0.9915
0.9958
0.9980
0.9991
0.9996
0.9998
0.9999
1.0000
CUMULATIVE POISSON PROBABILITY
20
x λ
Critical values for the product moment correlation coefficient, r
21/2%
5%
1%
2%
–
–
0.9877
0.9000
0.8054
0.7293
0.6694
0.6215
0.5822
0.5494
0.5214
0.4973
0.4762
0.4575
0.4409
0.4259
0.4124
0.4000
0.3887
0.3783
0.3687
0.3598
0.3515
0.3438
0.3365
0.3297
0.3233
0.3172
0.3115
0.3061
–
–
0.9969
0.9500
0.8783
0.8114
0.7545
0.7067
0.6664
0.6319
0.6021
0.5760
0.5529
0.5324
0.5140
0.4973
0.4821
0.4683
0.4555
0.4438
0.4329
0.4227
0.4132
0.4044
0.3961
0.3882
0.3809
0.3739
0.3673
0.3610
–
–
0.9995
0.9800
0.9343
0.8822
0.8329
0.7887
0.7498
0.7155
0.6851
0.6581
0.6339
0.6120
0.5923
0.5742
0.5577
0.5425
0.5285
0.5155
0.5034
0.4921
0.4815
0.4716
0.4622
0.4534
0.4451
0.4372
0.4297
0.4226
1
/2%
1%
–
–
0.9999
0.9900
0.9587
0.9172
0.8745
0.8343
0.7977
0.7646
0.7348
0.7079
0.6835
0.6614
0.6411
0.6226
0.6055
0.5897
0.5751
0.5614
0.5487
0.5368
0.5256
0.5151
0.5052
0.4958
0.4869
0.4785
0.4705
0.4629
1-Tail Test
2-Tail Test
n
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
5%
10%
21/2%
5%
1%
2%
0.3009
0.2960
0.2913
0.2869
0.2826
0.2785
0.2746
0.2709
0.2673
0.2638
0.2605
0.2573
0.2542
0.2512
0.2483
0.2455
0.2429
0.2403
0.2377
0.2353
0.2329
0.2306
0.2284
0.2262
0.2241
0.2221
0.2201
0.2181
0.2162
0.2144
0.3550
0.3494
0.3440
0.3388
0.3388
0.3291
0.3246
0.3202
0.3160
0.3120
0.3081
0.3044
0.3008
0.2973
0.2940
0.2907
0.2876
0.2845
0.2816
0.2787
0.2759
0.2732
0.2706
0.2681
0.2656
0.2632
0.2609
0.2586
0.2564
0.2542
0.4158
0.4093
0.4032
0.3972
0.3916
0.3862
0.3810
0.3760
0.3712
0.3665
0.3621
0.3578
0.3536
0.3496
0.3457
0.3420
0.3384
0.3348
0.3314
0.3281
0.3249
0.3218
0.3188
0.3158
0.3129
0.3102
0.3074
0.3048
0.3022
0.2997
1
/2%
1%
0.4556
0.4487
0.4421
0.4357
0.4926
0.4238
0.4182
0.4128
0.4076
0.4026
0.3978
0.3932
0.3887
0.3843
0.3801
0.3761
0.3721
0.3683
0.3646
0.3610
0.3575
0.3542
0.3509
0.3477
0.3445
0.3415
0.3385
0.3357
0.3328
0.3301
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
5%
10%
21/2%
5%
1%
2%
–
–
–
1.0000
0.9000
0.8286
0.7143
0.6429
0.6000
0.5636
0.5364
0.5035
0.4835
0.4637
0.4464
0.4294
0.4142
0.4014
0.3912
0.3805
0.3701
0.3608
0.3528
0.3443
0.3369
0.3306
0.3242
0.3180
0.3118
0.3063
–
–
–
–
1.0000
0.8857
0.7857
0.7381
0.7000
0.6485
0.6182
0.5874
0.5604
0.5385
0.5214
0.5029
0.4877
0.4716
0.4596
0.4466
0.4364
0.4252
0.4160
0.4070
0.3977
0.3901
0.3828
0.3755
0.3685
0.3624
–
–
–
–
1.0000
0.9429
0.8929
0.8333
0.7833
0.7455
0.7091
0.6783
0.6484
0.6264
0.6036
0.5824
0.5662
0.5501
0.5351
0.5218
0.5091
0.4975
0.4862
0.4757
0.4662
0.4571
0.4487
0.4401
0.4325
0.4251
1
/2%
1%
–
–
–
–
–
1.0000
0.9286
0.8810
0.8333
0.7939
0.7545
0.7273
0.7033
0.6791
0.6536
0.6353
0.6176
0.5996
0.5842
0.5699
0.5558
0.5438
0.5316
0.5209
0.5108
0.5009
0.4915
0.4828
0.4749
0.4670
1-Tail Test
2-Tail Test
n
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
5%
10%
21/2%
5%
1%
2%
0.3012
0.2962
0.2914
0.2871
0.2829
0.2788
0.2748
0.2710
0.2674
0.2640
0.2606
0.2574
0.2543
0.2513
0.2484
0.2456
0.2429
0.2403
0.2378
0.2353
0.2329
0.2307
0.2284
0.2262
0.2242
0.2221
0.2201
0.2181
0.2162
0.2144
0.3560
0.3504
0.3449
0.3396
0.3347
0.3300
0.3253
0.3209
0.3168
0.3128
0.3087
0.3051
0.3014
0.2978
0.2945
0.2913
0.2880
0.2850
0.2820
0.2791
0.2764
0.2736
0.2710
0.2685
0.2659
0.2636
0.2612
0.2589
0.2567
0.2545
0.4185
0.4117
0.4054
0.3995
0.3936
0.3882
0.3829
0.3778
0.3729
0.3681
0.3636
0.3594
0.3550
0.3511
0.3470
0.3433
0.3396
0.3361
0.3326
0.3293
0.3260
0.3228
0.3198
0.3168
0.3139
0.3111
0.3083
0.3057
0.3030
0.3005
1
/2%
1%
0.4593
0.4523
0.4455
0.4390
0.4328
0.4268
0.4211
0.4155
0.4103
0.4051
0.4002
0.3955
0.3908
0.3865
0.3882
0.3781
0.3741
0.3702
0.3664
0.3628
0.3592
0.3558
0.3524
0.3492
0.3460
0.3429
0.3400
0.3370
0.3342
0.3314
CRITICAL VALUES FOR CORRELATION COEFFICIENTS
21
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
5%
10%
Critical values for Spearman’s rank correlation coefficient, rs
The Normal distribution: values of Φ(z) = p
The Inverse Normal function: values of Φ–1(p) = z
The table gives the probability, p, of a
random variable distributed as N(0, 1)
being less than z.
N(0, 1)
p
z
(add)
.01
.02
.03
.04
.05
.06
.07
.08
.09
1 2
.5000
.5398
.5793
.6179
.6554
5040
5438
5832
6217
6591
5080
5478
5871
6255
6628
5120
5517
5910
6293
6664
5160
5557
5948
6331
6700
5199
5596
5987
6368
6736
5239
5636
6026
6406
6772
5279
5675
6064
6443
6808
5319
5714
6103
6480
6844
5359
5753
6141
6517
6879
4
4
4
4
4
8
8
8
8
7
3
4
5
6
7
8
9
12
12
12
11
11
16
16
15
15
14
20
20
19
19
18
24
24
23
23
22
28
28
27
26
25
32
32
31
30
29
36
35
35
34
32
0.5
0.6
0.7
0.8
0.9
.6915
.7257
.7580
.7881
.8159
6950
7291
7611
7910
8186
6985
7324
7642
7939
8212
7019
7357
7673
7967
8238
7054
7389
7704
7995
8264
7088
7422
7734
8023
8289
7123
7454
7764
8051
8315
7157
7486
7794
8078
8340
7190
7517
7823
8106
8365
7224
7549
7852
8133
8389
3
3
3
3
3
7 10 14 17 21 24
6 10 13 16 19 23
6 9 12 15 18 21
6 8 11 14 17 19
5 8 10 13 15 18
27
26
24
22
20
31
29
27
25
23
1.0
1.1
1.2
1.3
1.4
.8413
.8643
.8849
.9032
.9192
8438
8665
8869
9049
9207
8461
8686
8888
9066
9222
8485
8708
8907
9082
9236
8508
8729
8925
9099
9251
8531
8749
8944
9115
9265
8554
8770
8962
9131
9279
8577
8790
8980
9147
9292
8599
8810
8997
9162
9306
8621
8830
9015
9177
9319
2
2
2
2
1
5
4
4
3
3
7
6
6
5
4
9 12 14 16 18 21
8 10 12 14 16 19
7 9 11 13 15 16
6 8 10 11 13 14
6 7 8 10 11 13
1.5
1.6
1.7
1.8
1.9
.9332
.9452
.9554
.9641
.9713
9345
9463
9564
9649
9719
9357
9474
9573
9656
9726
9370
9484
9582
9664
9732
9382
9495
9591
9671
9738
9394
9505
9599
9678
9744
9406
9515
9608
9686
9750
9418
9525
9616
9693
9756
9429
9535
9625
9699
9761
9441
9545
9633
9706
9767
1
1
1
1
1
2
2
2
1
1
4
3
3
2
2
5
4
3
3
2
6
5
4
4
3
7
6
5
4
4
8 10 11
7 8 9
6 7 8
5 6 6
4 5 5
2.0
2.1
2.2
2.3
2.4
.9772
.9821
.9861
.9893
.9918
9778
9826
9864
9896
9920
9783
9830
9868
9898
9922
9788
9834
9871
9901
9925
9793
9838
9875
9904
9927
9798
9842
9878
9906
9929
9803
9846
9881
9909
9931
9808
9850
9884
9911
9932
9812
9854
9887
9913
9934
9817
9857
9890
9916
9936
0
0
0
0
0
1
1
1
1
0
1
1
1
1
1
2
2
1
1
1
2
2
2
1
1
3
2
2
2
1
3
3
2
2
1
2.5
2.6
2.7
2.8
2.9
.9938
.9953
.9965
.9974
.9981
9940
9955
9966
9975
9982
9941
9956
9967
9976
9982
9943
9957
9968
9977
9983
9945
9959
9969
9977
9984
9946
9960
9970
9978
9984
9948
9961
9971
9979
9985
9949
9962
9972
9979
9985
9951
9963
9973
9980
9986
9952
9964
9974
9981
9986
3.0
3.1
3.2
3.3
3.4
.9987
.9990
.9993
.9995
.9997
9987
9991
9993
9995
9997
9987
9991
9994
9996
9997
9988
9991
9994
9996
9997
9988
9992
9994
9996
9997
9989
9992
9994
9996
9997
9989
9992
9994
9996
9997
9989
9992
9995
9996
9997
9990
9993
9995
9996
9997
9990
9993
9995
9997
9998
differences
untrustworthy
4
3
3
2
2
4
4
3
2
2
.000
.0000
.0251
.0502
.0753
.1004
.1257
.1510
.1764
.2019
.2275
.2533
.2793
.3055
.3319
.3585
.3853
.4125
.4399
.4677
.4959
.5244
.5534
.5828
.6128
.6433
.6745
.7063
.7388
.7722
.8064
.8416
.8779
.9154
.9542
.9945
1.036
1.080
1.126
1.175
1.227
1.282
1.341
1.405
1.476
1.555
1.645
1.751
1.881
2.054
2.326
.001
.0025
.0276
.0527
.0778
.1030
.1282
.1535
.1789
.2045
.2301
.2559
.2819
.3081
.3345
.3611
.3880
.4152
.4427
.4705
.4987
.5273
.5563
.5858
.6158
.6464
.6776
.7095
.7421
.7756
.8099
.8452
.8816
.9192
.9581
.9986
1.041
1.085
1.131
1.180
1.232
1.287
1.347
1.412
1.483
1.563
1.655
1.762
1.896
2.075
2.366
.002
.0050
.0301
.0552
.0803
.1055
.1307
.1560
.1815
.2070
.2327
.2585
.2845
.3107
.3372
.3638
.3907
.4179
.4454
.4733
.5015
.5302
.5592
.5888
.6189
.6495
.6808
.7128
.7454
.7790
.8134
.8488
.8853
.9230
.9621
1.003
1.045
1.089
1.136
1.185
1.237
1.293
1.353
1.419
1.491
1.572
1.665
1.774
1.911
2.097
2.409
.003
.0075
.0326
.0577
.0828
.1080
.1332
.1586
.1840
.2096
.2353
.2611
.2871
.3134
.3398
.3665
.3934
.4207
.4482
.4761
.5044
.5330
.5622
.5918
.6219
.6526
.6840
.7160
.7488
.7824
.8169
.8524
.8890
.9269
.9661
1.007
1.049
1.094
1.141
1.190
1.243
1.299
1.360
1.426
1.499
1.581
1.675
1.787
1.927
2.120
2.457
.004
.0100
.0351
.0602
.0853
.1105
.1358
.1611
.1866
.2121
.2378
.2637
.2898
.3160
.3425
.3692
.3961
.4234
.4510
.4789
.5072
.5359
.5651
.5948
.6250
.6557
.6871
.7192
.7521
.7858
.8204
.8560
.8927
.9307
.9701
1.011
1.054
1.099
1.146
1.195
1.248
1.305
1.366
1.433
1.506
l.589
1.685
1.799
1.943
2.144
2.512
.005
.0125
.0376
.0627
.0878
.1130
.1383
.1637
.1891
.2147
.2404
.2663
.2924
.3186
.3451
.3719
.3989
.4261
.4538
.4817
.5101
.5388
.5681
.5978
.6280
.6588
.6903
.7225
.7554
.7892
.8239
.8596
.8965
.9346
.9741
1.015
1.058
1.103
1.150
1.200
l.254
1.311
1.372
1.440
1.514
1.598
1.695
1.812
1.960
2.170
2.576
.006
.0150
.0401
.0652
.0904
.1156
.1408
.1662
.1917
.2173
.2430
.2689
.2950
.3213
.3478
.3745
.4016
.4289
.4565
.4845
.5129
.5417
.5710
.6008
.6311
.6620
.6935
.7257
.7588
.7926
.8274
.8633
.9002
.9385
.9782
1.019
1.063
1.108
1.155
1.206
1.259
1.317
1.379
1.447
1.522
1.607
1.706
1.825
1.977
2.197
2.652
.007
.0175
.0426
.0677
.0929
.1181
.1434
.1687
.1942
.2198
.2456
.2715
.2976
.3239
.3505
.3772
.4043
.4316
.4593
.4874
.5158
.5446
.5740
.6038
.6341
.6651
.6967
.7290
.7621
.7961
.8310
.8669
.9040
.9424
.9822
1.024
1.067
1.112
1.160
1.211
1.265
1.323
1.385
1.454
1.530
1.616
1.717
1.838
1.995
2.226
2.748
.008
.0201
.0451
.0702
.0954
.1206
.1459
.1713
.1968
.2224
.2482
.2741
.3002
.3266
.3531
.3799
.4070
.4344
.4621
.4902
.5187
.5476
.5769
.6068
.6372
.6682
.6999
.7323
.7655
.7995
.8345
.8705
.9078
.9463
.9863
1.028
1.071
1.117
1.165
1.216
1.270
1.329
1.392
1.461
1.538
1.626
1.728
1.852
2.014
2.257
2.878
.009
.0226
.0476
.0728
.0979
.1231
.1484
.1738
.1993
.2250
.2508
.2767
.3029
.3292
.3558
.3826
.4097
.4372
.4649
.4930
.5215
.5505
.5799
.6098
.6403
.6713
.7031
.7356
.7688
.8030
.838l
.8742
.9116
.9502
.9904
1.032
1.076
1.122
1.170
1.221
1.276
1.335
1.398
1.468
1.546
1.635
1.739
1.866
2.034
2.290
3.090
THE NORMAL DISTRIBUTION AND ITS INVERSE
z
22
.00
0.0
0.1
0.2
0.3
0.4
p
.50
.51
.52
.53
.54
.55
.56
.57
.58
.59
.60
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
.90
.91
.92
.93
.94
.95
.96
.97
.98
.99
Percentage points of the χ2 (chi-squared) distribution
p%
p%
χ2
χ2
p%
97.5
95
90
10
5.0
2.5
1.0
0.5
.0001
.0201
0.115
0.297
0.554
0.872
1.239
1.646
2.088
2.558
3.053
3.571
4.107
4.660
5.229
5.812
6.408
7.015
7.633
8.260
8.897
9.542
10.20
10.86
11.52
12.20
12.88
13.56
14.26
14.95
18.51
22.16
29.71
70.06
.0010
.0506
0.216
0.484
0.831
1.237
1.690
2.180
2.700
3.247
3.816
4.404
5.009
5.629
6.262
6.908
7.564
8.231
8.907
9.591
10.28
10.98
11.69
12.40
13.12
13.84
14.57
15.31
16.05
16.79
20.57
24.43
32.36
74.22
.0039
0.103
0.352
0.711
1.145
1.635
2.167
2.733
3.325
3.940
4.575
5.226
5.892
6.571
7.261
7.962
8.672
9.390
10.12
10.85
11.59
12.34
13.09
13.85
14.61
15.38
16.15
16.93
17.71
18.49
22.47
26.51
34.76
77.93
.0158
0.211
0.584
1.064
1.610
2.204
2.833
3.490
4.168
4.865
5.578
6.304
7.042
7.790
8.547
9.312
10.09
10.86
11.65
12.44
13.24
14.04
14.85
15.66
16.47
17.29
18.11
18.94
19.77
20.60
24.80
29.05
37.69
82.36
2.706
4.605
6.251
7.779
9.236
10.64
12.02
13.36
14.68
15.99
17.28
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56
36.74
37.92
39.09
40.26
46.06
51.81
63.17
118.5
3.841
5.991
7.815
9.488
11.07
12.59
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
32.67
33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
49.80
55.76
67.50
124.3
5.024
7.378
9.348
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
53.20
59.34
71.42
129.6
6.635
9.210
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
24.72
26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
57.34
63.69
76.15
135.8
7.879
10.60
12.84
14.86
16.75
18.55
20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.64
50.99
52.34
53.67
60.27
66.77
79.49
140.2
1/2
p%
1/2
p%
t
p%
v=1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
20
30
50
100
∞
10
5
2
1
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.725
1.697
1.676
1.660
1.645
12.71
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.086
2.042
2.009
1.984
1.960
31.82
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.528
2.457
2.403
2.364
2.326
63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.845
2.750
2.678
2.626
2.576
2
99
PERCENTAGE POINTS OF χ AND t – DISTRIBUTIONS
23
v=1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
35
40
50
100
Percentage points of the t-distribution
= Percentage points of the
Normal distribution N(0, 1)
21/2% points of the F-distribution
5% points of the F-distribution
v2
12
24
∞
v1
1
2
3
4
5
6
7
8
10
12
24
∞
238.9
19.4
8.85
6.04
241.9
19.4
8.79
5.96
243.9
19.4
8.74
5.91
249.0
19.5
8.64
5.77
254.3
19.5
8.53
5.63
1
2
3
4
648
38.5
17.4
12.22
800
39.0
16.0
10.65
864
39.2
15.4
9.98
900
39.2
15.1
9.60
922
39.3
14.9
9.36
937
39.3
14.7
9.20
948
39.4
14.6
9.07
957
39.4
14.5
8.98
969
39.4
14.4
8.84
977
39.4
14.3
8.75
997
39.5
14.1
8.51
1018
39.5
13.9
8.26
4.88
4.21
3.79
3.50
3.29
4.82
4.15
3.73
3.44
3.23
4.74
4.06
3.64
3.35
3.14
4.68
4.00
3.57
3.28
3.07
4.53
3.84
3.41
3.12
2.90
4.36
3.67
3.23
2.93
2.71
5
6
7
8
9
10.01
8.81
8.07
7.57
7.21
8.43
7.26
6.54
6.06
5.71
7.76
6.60
5.89
5.42
5.08
7.39
6.23
5.52
5.05
4.72
7.15
5.99
5.29
4.82
4.48
6.98
5.82
5.12
4.65
4.32
6.85
5.70
4.99
4.53
4.20
6.76
5.60
4.90
4.43
4.10
6.62
5.46
4.76
4.30
3.96
6.52
5.37
4.67
4.20
3.87
6.28
5.12
4.42
3.95
3.61
6.02
4.85
4.14
3.67
3.33
3.22
3.09
3.00
2.92
2.85
3.14
3.01
2.91
2.83
2.76
3.07
2.95
2.85
2.77
2.70
2.98
2.85
2.75
2.67
2.60
2.91
2.79
2.69
2.60
2.53
2.74
2.61
2.51
2.42
2.35
2.54
2.40
2.30
2.21
2.13
10
11
12
13
14
6.94
6.72
6.55
6.41
6.30
5.46
5.26
5.10
4.97
4.86
4.83
4.63
4.47
4.35
4.24
4.47
4.28
4.12
4.00
3.89
4.24
4.04
3.89
3.77
3.66
4.07
3.88
3.73
3.60
3.50
3.95
3.76
3.61
3.48
3.38
3.85
3.66
3.51
3.39
3.29
3.72
3.53
3.37
3.25
3.15
3.62
3.43
3.28
3.15
3.05
3.37
3.17
3.02
2.89
2.79
3.08
2.88
2.72
2.60
2.49
2.90
2.85
2.81
2.77
2.74
2.79
2.74
2.70
2.66
2.63
2.71
2.66
2.61
2.58
2.54
2.64
2.59
2.55
2.51
2.48
2.54
2.49
2.45
2.41
2.38
2.48
2.42
2.38
2.34
2.31
2.29
2.24
2.19
2.15
2.11
2.07
2.01
1.96
1.92
1.88
15
16
17
18
19
6.20
6.12
6.04
5.98
5.92
4.76
4.69
4.62
4.56
4.51
4.15
4.08
4.01
3.95
3.90
3.80
3.73
3.66
3.61
3.56
3.58
3.50
3.44
3.38
3.33
3.41
3.34
3.28
3.22
3.17
3.29
3.22
3.16
3.10
3.05
3.20
3.12
3.06
3.01
2.96
3.06
2.99
2.92
2.87
2.82
2.96
2.89
2.82
2.77
2.72
2.70
2.63
2.56
2.50
2.45
2.40
2.32
2.25
2.19
2.13
2.87
2.84
2.82
2.80
2.78
2.71
2.68
2.66
2.64
2.62
2.60
2.57
2.55
2.53
2.51
2.51
2.49
2.46
2.44
2.42
2.45
2.42
2.40
2.37
2.36
2.35
2.32
2.30
2.27
2.25
2.28
2.25
2.23
2.20
2.18
2.08
2.05
2.03
2.00
1.98
1.84
1.81
1.78
1.76
1.73
20
21
22
23
24
5.87
5.83
5.79
5.75
5.72
4.46
4.42
4.38
4.35
4.32
3.86
3.82
3.78
3.75
3.72
3.51
3.48
3.44
3.41
3.38
3.29
3.25
3.22
3.18
3.15
3.13
3.09
3.05
3.02
2.99
3.01
2.97
2.93
2.90
2.87
2.91
2.87
2.84
2.81
2.78
2.77
2.73
2.70
2.67
2.64
2.68
2.64
2.60
2.57
2.54
2.41
2.37
2.33
2.30
2.27
2.09
2.04
2.00
1.97
1.94
2.99
2.98
2.96
2.95
2.93
2.76
2.74
2.73
2.71
2.70
2.60
2.59
2.57
2.56
2.55
2.49
2.47
2.46
2.45
2.43
2.40
2.39
2.37
2.36
2.35
2.34
2.32
2.31
2.29
2.28
2.24
2.22
2.20
2.19
2.18
2.16
2.15
2.13
2.12
2.10
1.96
1.95
1.93
1.91
1.90
1.71
1.69
1.67
1.65
1.64
25
26
27
28
29
5.69
5.66
5.63
5.61
5.59
4.29
4.27
4.24
4.22
4.20
3.69
3.67
3.65
3.63
3.61
3.35
3.33
3.31
3.29
3.27
3.13
3.10
3.08
3.06
3.04
2.97
2.94
2.92
2.90
2.88
2.85
2.82
2.80
2.78
2.76
2.75
2.73
2.71
2.69
2.67
2.61
2.59
2.57
2.55
2.53
2.51
2.49
2.47
2.45
2.43
2.24
2.22
2.19
2.17
2.15
1.91
1.88
1.85
1.83
1.81
3.32
3.29
3.28
3.26
3.24
2.92
2.90
2.88
2.87
2.85
2.69
2.67
2.65
2.63
2.62
2.53
2.51
2.49
2.48
2.46
2.42
2.40
2.38
2.36
2.35
2.33
2.31
2.29
2.28
2.26
2.27
2.24
2.23
2.21
2.19
2.16
2.14
2.12
2.11
2.09
2.09
2.07
2.05
2.03
2.02
1.89
1.86
1.84
1.82
1.81
1.62
1.59
1.57
1.55
1.53
30
32
34
36
38
5.57
5.53
5.50
5.47
5.45
4.18
4.15
4.12
4.09
4.07
3.59
3.56
3.53
3.51
3.48
3.25
3.22
3.19
3.17
3.15
3.03
3.00
2.97
2.94
2.92
2.87
2.84
2.81
2.79
2.76
2.75
2.72
2.69
2.66
2.64
2.65
2.62
2.59
2.57
2.55
2.51
2.48
2.45
2.43
2.41
2.41
2.38
2.35
2.33
2.31
2.14
2.10
2.08
2.05
2.03
1.79
1.75
1.72
1.69
1.66
3.23
3.15
3.07
3.00
2.84
2.76
2.68
2.60
2.61
2.53
2.45
2.37
2.45
2.37
2.29
2.21
2.34
2.25
2.18
2.10
2.25
2.17
2.09
2.01
2.18
2.10
2.02
1.94
2.08
1.99
1.91
1.83
2.00
1.92
1.83
1.75
1.79
1.70
1.61
1.52
1.51
1.39
1.25
1.00
40
60
120
∞
5.42
5.29
5.15
5.02
4.05
3.93
3.80
3.69
3.46
3.34
3.23
3.12
3.13
3.01
2.89
2.79
2.90
2.79
2.67
2.57
2.74
2.63
2.52
2.41
2.62
2.51
2.39
2.29
2.53
2.41
2.30
2.19
2.39
2.27
2.16
2.05
2.29
2.17
2.05
1.94
2.01
1.88
1.76
1.64
1.64
1.48
1.31
1.00
1
2
3
4
5
6
7
8
1
2
3
4
161.4
18.5
10.13
7.71
199.5
19.0
9.55
6.94
215.7
19.2
9.28
6.59
224.6
19.2
9.12
6.39
230.2
19.3
9.01
6.26
234.0
19.3
8.94
6.16
236.8
19.4
8.89
6.09
5
6
7
8
9
6.61
5.99
5.59
5.32
5.12
5.79
5.14
4.74
4.46
4.26
5.41
4.76
4.35
4.07
3.86
5.19
4.53
4.12
3.84
3.63
5.05
4.39
3.97
3.69
3.48
4.95
4.28
3.87
3.58
3.37
10
11
12
13
14
4.96
4.84
4.75
4.67
4.60
4.10
3.98
3.89
3.81
3.74
3.71
3.59
3.49
3.41
3.34
3.48
3.36
3.26
3.18
3.11
3.33
3.20
3.11
3.03
2.96
15
16
17
18
19
4.54
4.49
4.45
4.41
4.38
3.68
3.63
3.59
3.55
3.52
3.29
3.24
3.20
3.16
3.13
3.06
3.01
2.96
2.93
2.90
20
21
22
23
24
4.35
4.32
4.30
4.28
4.26
3.49
3.47
3.44
3.42
3.40
3.10
3.07
3.05
3.03
3.01
25
26
27
28
29
4.24
4.23
4.21
4.20
4.18
3.39
3.37
3.35
3.34
3.33
30
32
34
36
38
4.17
4.15
4.13
4.11
4.10
40
60
120
∞
4.08
4.00
3.92
3.84
v2
CRITICAL VALUES FOR F – TEST
24
10
v1
1% points of the F-distribution
v2
0.1% points of the F-distribution
1
2
3
4
5
6
7
8
10
12
24
∞
v1
1
2
3
4
5
6
7
8
10
12
24
∞
1
2
3
4
4052
98.5
34.1
21.2
5000
99.0
30.8
18.0
5403
99.2
29.5
16.7
5625
99.2
28.7
16.0
5764
99.3
28.2
15.5
5859
99.3
27.9
15.2
5928
99.4
27.7
15.0
5981
99.4
27.5
14.8
6056
99.4
27.2
14.5
6106
99.4
27.1
14.4
6235
99.5
26.6
13.9
6366
99.5
26.1
13.5
1
2
3
4
4053
998.5.
167.0
74.14
5000
999.0
148.5
61.25
5404
999.2
141.1
56.18
5625
999.2
137.1
53.44
5764
999.3
134.6
51.71
5859
999.3
132.8
50.53
5929
999.4
131.5
49.66
5981
999.4
130.6
49.00
6056
999.4
129.2
48.05
6107
999.4
128.3
47.41
6235
999.5
125.9
45.77
6366
999 5
123.5
44.05
5
6
7
8
9
16.26
13.74
12.25
11.26
10.56
13.27
10.92
9.55
8.65
8.02
12.06
9.78
8.45
7.59
6.99
11.39
9.15
7.85
7.01
6.42
10.97
8.75
7.46
6.63
6.06
10.67
8.47
7.19
6.37
5.80
10.46
8.26
6.99
6.18
5.61
10.29
8.10
6.84
6.03
5.47
10.05
7.87
6.62
5.81
5.26
9.89
7.72
6.47
5.67
5.11
9.47
7.31
6.07
5.28
4.73
9.02
6.88
5.65
4.86
4.31
5
6
7
8
9
47.18
35.51
29.25
25.42
22.86
37.12
27.00
21.69
18.49
16.39
33.20
23.70
18.77
15.83
13.90
31.09
21.92
17.20
14.39
12.56
29.75
20.80
16.21
13.48
11.71
28.83
20.03
15.52
12.86
11.13
28.16
19.46
15.02
12.40
10.69
27.65
19.03
14.63
12.05
10.37
26.92
18.41
14.08
11.54
9.87
26.42
17.99
13.71
11.19
9.57
25.14
16.90
12.73
10.30
8.72
23.79
15.75
11.70
9.34
7.81
10
11
12
13
14
10.04
9.65
9.33
9.07
8.86
7.56
7.21
6.93
6.70
6.51
6.55
6.22
5.95
5.74
5.56
5.99
5.67
5.41
5.21
5.04
5.64
5.32
5.06
4.86
4.70
5.39
5.07
4.82
4.62
4.46
5.20
4.89
4.64
4.44
4.28
5.06
4.74
4.50
4.30
4.14
4.85
4.54
4.30
4.10
3.94
4.71
4.40
4.16
3.96
3.80
4.33
4.02
3.78
3.59
3.43
3.91
3.60
3.36
3.17
3.00
10
11
12
13
14
21.04
19.69
18.64
17.82
17.14
14.91
13.81
12.97
12.31
11.78
12.55
11.56
10.80
10.21
9.73
11.28
10.35
9.63
9.07
8.62
10.48
9.58
8.89
8.35
7.92
9.93
9.05
8.38
7.86
7.44
9.52
8.66
8.00
7.49
7.08
9.20
8.35
7.71
7.21
6.80
8.74
7.92
7.29
6.80
6.40
8.44
7.63
7.00
6.52
6.13
7.64
6.85
6.25
5.78
5.41
6.76
6.00
5.42
4.97
4.60
15
16
17
18
19
8.68
8.53
8.40
8.29
8.18
6.36
6.23
6.11
6.01
5.93
5.42
5.29
5.18
5.09
5.01
4.89
4.77
4.67
4.58
4.50
4.56
4.44
4.34
4.25
4.17
4.32
4.20
4.10
4.01
3.94
4.14
4.03
3.93
3.84
3.77
4.00
3.89
3.79
3.71
3.63
3.80
3.69
3.59
3.51
3.43
3.67
3.55
3.46
3.37
3.30
3.29
3.18
3.08
3.00
2.92
2.87
2.75
2.65
2.57
2.49
15
16
17
18
19
16.59
16.12
15.72
15.38
15.08
11.34
10.97
10.66
10.39
10.16
9.34
9.01
8.73
8.49
8.28
8.25
7.94
7.68
7.46
7.27
7.57
7.27
7.02
6.81
6.62
7.09
6.80
6.56
6.35
6.18
6.74
6.46
6.22
6.02
5.85
6.47
6.19
5.96
5.76
5.59
6.08
5.81
5.58
5.39
5.22
5.81
5.55
5.32
5.13
4.97
5.10
4.85
4.63
4.45
4.29
4.31
4.06
3.85
3.67
3.51
20
21
22
23
24
8.10
8.02
7.95
7.88
7.82
5.85
5.78
5.72
5.66
5.61
4.94
4.87
4.82
4.76
4.72
4.43
4.37
4.31
4.26
4.22
4.10
4.04
3.99
3.94
3.90
3.87
3.81
3.76
3.71
3.67
3.70
3.64
3.59
3.54
3.50
3.56
3.51
3.45
3.41
3.36
3.37
3.31
3.26
3.21
3.17
3.23
3.17
3.12
3.07
3.03
2.86
2.80
2.75
2.70
2.66
2.42
2.36
2.31
2.26
2.21
20
21
22
23
24
14.82
14.59
14.38
14.19
14.03
9.95
9.77
9.61
9.47
9.34
8.10
7.94
7.80
7.67
7.55
7.10
6.95
6.81
6.70
6.59
6.46
6.32
6.19
6.08
5.98
6.02
5.88
5.76
5.65
5.55
5.69
5.56
5.44
5.33
5.23
5.44
5.31
5.19
5.09
4.99
5.08
4.95
4.83
4.73
4.64
4.82
4.70
4.58
4.48
4.39
4.15
4.03
3.92
3.82
3.74
3.38
3.26
3.15
3.05
2.97
25
26
27
28
29
7.77
7.72
7.68
7.64
7.60
5.57
5.53
5.49
5.45
5.42
4.68
4.64
4.60
4.57
4.54
4.18
4.14
4.11
4.07
4.04
3.86
3.82
3.78
3.75
3.73
3.63
3.59
3.56
3.53
3.50
3.46
3.42
3.39
3.36
3.33
3.32
3.29
3.26
3.23
3.20
3.13
3.09
3.06
3.03
3.00
2.99
2.96
2.93
2.90
2.87
2.62
2.58
2.55
2.52
2.49
2.17
2.13
2.10
2.06
2.03
25
26
27
28
29
13.88
13.74
13.61
13.50
13.39
9.22
9.12
9.02
8.93
8.85
7.45
7.36
7.27
7.19
7.12
6.49
6.41
6.33
6.25
6.19
5.89
5.80
5.73
5.66
5.59
5.46
5.38
5.31
5.24
5.18
5.15
5.07
5.00
4.93
4.87
4.91
4.83
4.76
4.69
4.64
4.56
4.48
4.41
4.35
4.29
4.31
4.24
4.17
4.11
4.05
3.66
3.59
3.52
3.46
3.41
2.89
2.82
2.75
2.69
2.64
30
32
34
36
38
7.56
7.50
7.45
7.40
7.35
5.39
5.34
5.29
5.25
5.21
4.51
4.46
4.42
4.38
4.34
4.02
3.97
3.93
3.89
3.86
3.70
3.65
3.61
3.58
3.54
3.47
3.43
3.39
3.35
3.32
3.30
3.26
3.22
3.18
3.15
3.17
3.13
3.09
3.05
3.02
2.98
2.93
2.90
2.86
2.83
2.84
2.80
2.76
2.72
2.69
2.47
2.42
2.38
2.35
2.32
2.01
1.96
1.91
1.87
1.84
30
32
34
36
38
13.29
13.12
12.97
12.83
12.71
8.77
8.64
8.52
8.42
8.33
7.05
6.94
6.83
6.74
6.66
6.12
6.01
5.92
5.84
5.76
5.53
5.43
5.34
5.26
5.19
5.12
5.02
4.93
4.86
4.79
4.82
4.72
4.63
4.56
4.49
4.58
4.48
4.40
4.33
4.26
4.24
4.14
4.06
3.99
3.93
4.00
3.91
3.83
3.76
3.70
3.36
3.27
3.19
3.12
3.06
2.59
2.50
2.42
2.35
2.29
40
60
120
∞
7.31
7.08
6.85
6.63
5.18
4.98
4.79
4.61
4.31
4.13
3.95
3.78
3.83
3.65
3.48
3.32
3.51
3.34
3.17
3.02
3.29
3.12
2.96
2.80
3.12
2.95
2.79
2.64
2.99
2.82
2.66
2.51
2.80
2.63
2.47
2.32
2.66
2.50
2.34
2.18
2.29
2.12
1.95
1.79
1.80
1.60
1.38
1.00
40
60
120
∞
12.61
11.97
11.38
10.83
8.25
7.77
7.32
6.91
6.59
6.17
5.78
5.42
5.70
5.31
4.95
4.62
5.13
4.76
4.42
4.10
4.73
4.37
4.04
3.74
4.44
4.09
3.77
3.47
4.21
3.86
3.55
3.27
3.87
3.54
3.24
2.96
3.64
3.32
3.02
2.74
3.01
2.69
2.40
2.13
2.23
1.89
1.54
1.00
v2
CRITICAL VALUES FOR F – TEST
25
v1
Critical Values for the Mann-Whitney Test
1 – tail
2 – tail
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
3
4
5
6
7
8
9
10
11
12
13
–
–
–
0
0
0
1
1
1
1
2
2
3
3
3
3
4
4
4
5
5
5
6
6
0
0
1
2
2
3
4
4
5
5
6
–
–
–
–
–
–
0
0
0
0
1
1
1
1
1
2
2
2
2
3
3
3
3
3
–
–
0
1
1
2
2
3
3
4
4
–
–
–
–
–
–
–
–
–
–
–
0
0
0
0
0
0
1
1
1
1
1
1
1
–
–
–
–
0
0
1
1
1
2
2
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
0
0
0
0
0
0
0
–
–
–
–
–
–
0
0
0
1
1
1 – tail
2 – tail
m
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
n
14
15
16
17
18
19
20
21
22
23
24
25
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
7
7
8
9
9
10
11
11
12
13
13
14
1
2
3
4
5
6
7
8
9
10
11
12
14
15
16
17
18
19
20
21
22
23
5
5
6
6
7
7
8
8
9
9
10
10
0
1
2
3
4
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
17
18
2 1
3 2
3 2
4 2
4 2
4 3
5 3
5 3
6 4
6 4
6 4
7 5
– –
0 –
1 0
1 0
2 1
3 1
3 2
4 2
5 3
5 3
6 4
7 5
7 5
8 6
9 6
9 7
10 8
11 8
11 9
12 9
13 10
13 10
The critical values in these tables are for
the Mann-Whitney test statistic, T.
Critical values for the Wilcoxon test
statistic, W, may be derived by adding
1–
m(m + 1) (where m is the size of
2
the sample from which the rank sum
has been obtained). These values are
tabulated on pages 28 and 29.
1 – tail
2 – tail
m
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
n
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
6
7
8
9
5% 21/2% 1% 1/2%
10% 5% 2% 1%
4
5
6
8
9
11
12
13
15
16
18
19
20
22
23
25
26
28
29
30
32
7
8
10
12
2
3
5
6
7
8
9
11
12
13
14
15
17
18
19
20
22
23
24
25
27
5
6
8
10
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
3
4
6
7
0
1
1
2
3
4
5
6
7
7
8
9
10
11
12
13
14
14
15
16
17
2
3
4
5
1 – tail
2 – tail
m
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
n
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
14
16
17
19
21
23
25
26
28
30
32
34
36
37
39
41
11
13
15
17
19
21
24
26
28
30
33
35
37
39
41
44
46
48
50
11
13
14
16
17
19
21
22
24
25
27
29
30
32
33
35
8
10
12
14
16
18
20
22
24
26
28
30
32
34
36
38
40
42
44
8
9
11
12
13
15
16
18
19
20
22
23
24
26
27
29
6
7
9
11
12
14
16
17
19
21
23
24
26
28
30
31
33
35
36
6
7
9
10
11
12
13
15
16
17
18
19
21
22
23
24
4
6
7
9
10
12
13
15
16
18
19
21
22
24
25
27
29
30
32
1 – tail
2 – tail
m
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
n
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
15
18
20
23
26
28
31
33
36
39
41
44
47
49
52
54
57
60
21
24
27
30
33
36
39
42
45
48
51
54
57
60
63
66
69
13
15
17
19
22
24
26
29
31
34
36
38
41
43
45
48
50
53
17
20
23
26
28
31
34
37
39
42
45
48
50
53
56
59
62
9
11
13
15
17
20
22
24
26
28
30
32
34
36
38
40
42
45
14
16
18
21
23
26
28
31
33
35
38
40
43
45
48
50
53
7
9
11
13
15
17
18
20
22
24
26
28
30
32
34
35
37
39
11
13
16
18
20
22
24
27
29
31
33
36
38
40
43
45
47
CRITICAL VALUES FOR THE MANN-WHITNEY TEST
26
m
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
5% 21/2% 1% 1/2%
10% 5% 2% 1%
1 – tail
2 – tail
n
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
27
31
34
37
41
44
48
51
55
58
62
65
68
72
75
79
34
38
42
46
50
54
57
61
65
69
73
77
81
85
89
23
26
29
33
36
39
42
45
48
52
55
58
61
64
67
71
30
33
37
40
44
47
51
55
58
62
65
69
73
76
80
19
22
24
27
30
33
36
38
41
44
47
50
53
55
58
61
25
28
31
34
37
41
44
47
50
53
57
60
63
66
70
16
18
21
24
26
29
31
34
37
39
42
44
47
50
52
55
21
24
27
30
33
36
39
42
45
48
51
54
57
60
63
1 – tail
2 – tail
m
12
12
12
12
12
12
12
12
12
12
12
12
12
12
13
13
13
13
13
13
13
13
13
13
13
13
13
n
12
13
14
15
16
17
18
19
20
21
22
23
24
25
13
14
15
16
17
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
42
47
51
55
60
64
68
72
77
81
85
90
94
98
51
56
61
65
70
75
80
84
89
94
98
103
108
37
41
45
49
53
57
61
65
69
73
77
81
85
89
45
50
54
59
63
67
72
76
80
85
89
94
98
31
35
38
42
46
49
53
56
60
64
67
71
75
78
39
43
47
51
55
59
63
67
71
75
79
83
87
27
31
34
37
41
44
47
51
54
58
61
64
68
71
34
38
42
45
49
53
57
60
64
68
72
75
79
1 – tail
2 – tail
m
14
14
14
14
14
14
14
14
14
14
14
14
15
15
15
15
15
15
15
15
15
15
15
16
16
16
n
14
15
16
17
18
19
20
21
22
23
24
25
15
16
17
18
19
20
21
22
23
24
25
16
17
18
5% 21/2% 1% 1/2%
10% 5% 2% 1%
61
66
71
77
82
87
92
97
102
107
113
118
72
77
83
88
94
100
105
111
116
122
128
83
89
95
55 47
59 51
64 56
69 60
74 65
78 69
83 73
88 78
93 82
98 87
102 91
107 95
64 56
70 61
75 66
80 70
85 75
90 80
96 85
101 90
106 94
111 99
117 104
75 66
81 71
86 76
42
46
50
54
58
63
67
71
75
79
83
87
51
55
60
64
69
73
78
82
87
91
96
60
65
70
1 – tail
2 – tail
m
16
16
16
16
16
16
16
17
17
17
17
17
17
17
17
17
18
18
18
18
18
18
18
18
n
19
20
21
22
23
24
25
17
18
19
20
21
22
23
24
25
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
101
107
113
119
125
131
137
96
102
109
115
121
128
134
141
147
109
116
123
130
136
143
150
157
92
98
103
109
115
120
126
87
93
99
105
111
117
123
129
135
99
106
112
119
125
132
138
145
82
87
92
97
102
108
113
77
82
88
93
99
105
110
116
122
88
94
100
106
112
118
124
130
74
79
84
89
94
99
104
70
75
81
86
91
96
102
107
112
81
87
92
98
104
109
115
121
For larger values of m, n it is usually adequate to use a Normal approximation with continuity correction,
with mean
1
–
2
mn and variance
1
–– mn(m
12
+ n + 1).
1 – tail
2 – tail
5% 21/2% 1% 1/2%
10% 5% 2% 1%
m
19
19
19
19
19
19
19
20
20
20
20
20
20
21
21
21
21
21
22
22
22
22
23
23
23
24
24
25
123
130
138
145
152
160
167
138
146
154
161
169
177
154
162
170
179
187
171
179
188
197
189
198
207
207
217
227
n
19
20
21
22
23
24
25
20
21
22
23
24
25
21
22
23
24
25
22
23
24
25
23
24
25
24
25
25
113
119
126
133
140
147
154
127
134
141
149
156
163
142
150
157
165
173
158
166
174
182
175
183
192
192
201
211
101
107
113
120
126
133
139
114
121
127
134
141
148
128
135
142
150
157
143
150
158
166
158
167
175
175
184
192
93
99
105
111
117
123
129
105
112
118
125
131
138
118
125
132
139
146
133
140
147
155
148
155
163
164
172
180
CRITICAL VALUES FOR THE MANN-WHITNEY TEST
27
m
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
5% 21/2% 1% 1/2%
10% 5% 2% 1%
Critical Values for the Wilcoxon Rank Sum 2-Sample Test
1 – tail
2 – tail
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
3
4
5
6
7
8
9
10
11
12
13
– –
– –
– –
3 –
3 –
3 –
4 3
4 3
4 3
4 3
5 4
5 4
6 4
6 4
6 4
6 5
7 5
7 5
7 5
8 6
8 6
8 6
9 6
9 6
6 –
6 –
7 6
8 7
8 7
9 8
10 8
10 9
11 9
11 10
12 10
–
–
–
–
–
–
–
–
–
–
–
3
3
3
3
3
3
4
4
4
4
4
4
4
–
–
–
–
6
6
7
7
7
8
8
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
3
3
3
3
3
3
3
–
–
–
–
–
–
6
6
6
7
7
1 – tail
2 – tail
m
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
n
14
15
16
17
18
19
20
21
22
23
24
25
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
13
13
14
15
15
16
17
17
18
19
19
20
11
12
13
14
15
16
17
18
19
20
21
22
24
25
26
27
28
29
30
31
32
33
11
11
12
12
13
13
14
14
15
15
16
16
10
11
12
13
14
14
15
16
17
18
19
20
21
21
22
23
24
25
26
27
27
28
8
9
9
10
10
10
11
11
12
12
12
13
–
10
11
11
12
13
13
14
15
15
16
17
17
18
19
19
20
21
21
22
23
23
7
8
8
8
8
9
9
9
10
10
10
11
–
–
10
10
11
11
12
12
13
13
14
15
15
16
16
17
18
18
19
19
20
20
The critical values in these tables are
for the Wilcoxon Rank Sum 2-sample
test statistic, W. Critical values for
the Mann-Whitney test statistic, T, may
be derived by subtracting 1–2 m(m + 1)
(where m is the size of the sample from
which the rank sum has been obtained).
1 – tail
2 – tail
m
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
n
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
6
7
8
9
5% 21/2% 1% 1/2%
10% 5% 2% 1%
19
20
21
23
24
26
27
28
30
31
33
34
35
37
38
40
41
43
44
45
47
28
29
31
33
17
18
20
21
22
23
24
26
27
28
29
30
32
33
34
35
37
38
39
40
42
26
27
29
31
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
24
25
27
28
15
16
16
17
18
19
20
21
22
22
23
24
25
26
27
28
29
29
30
31
32
23
24
25
26
1 – tail
2 – tail
m
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
n
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
35
37
38
40
42
44
46
47
49
51
53
55
57
58
60
62
39
41
43
45
47
49
52
54
56
58
61
63
65
67
69
72
74
76
78
32
34
35
37
38
40
42
43
45
46
48
50
51
53
54
56
36
38
40
42
44
46
48
50
52
54
56
58
60
62
64
66
68
70
72
29
30
32
33
34
36
37
39
40
41
43
44
45
47
48
50
34
35
37
39
40
42
44
45
47
49
51
52
54
56
58
59
61
63
64
27
28
30
31
32
33
34
36
37
38
39
40
42
43
44
45
32
34
35
37
38
40
41
43
44
46
47
49
50
52
53
55
57
58
60
1 – tail
2 – tail
m
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
n
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
51
54
56
59
62
64
67
69
72
75
77
80
83
85
88
90
93
96
66
69
72
75
78
81
84
87
90
93
96
99
102
105
108
111
114
49
51
53
55
58
60
62
65
67
70
72
74
77
79
81
84
86
89
62
65
68
71
73
76
79
82
84
87
90
93
95
98
101
104
107
45
47
49
51
53
56
58
60
62
64
66
68
70
72
74
76
78
81
59
61
63
66
68
71
73
76
78
80
83
85
88
90
93
95
98
43
45
47
49
51
53
54
56
58
60
62
64
66
68
70
71
73
75
56
58
61
63
65
67
69
72
74
76
78
81
83
85
88
90
92
CRITICAL VALUES FOR THE WILCOXON RANK SUM 2-SAMPLE TEST
28
m
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
5% 21/2% 1% 1/2%
10% 5% 2% 1%
1 – tail
2 – tail
n
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
82
86
89
92
96
99
103
106
110
113
117
120
123
127
130
134
100
104
108
112
116
120
123
127
131
135
139
143
147
151
155
78
81
84
88
91
94
97
100
103
107
110
113
116
119
122
126
96
99
103
106
110
113
117
121
124
128
131
135
139
142
146
74
77
79
82
85
88
91
93
96
99
102
105
108
110
113
116
91
94
97
100
103
107
110
113
116
119
123
126
129
132
136
1 – tail
2 – tail
m
12
12
12
12
12
12
12
12
12
12
12
12
12
12
13
13
13
13
13
13
13
13
13
13
13
13
13
n
12
13
14
15
16
17
18
19
20
21
22
23
24
25
13
14
15
16
17
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
1 – tail
2 – tail
m
14
14
14
14
14
14
14
14
14
14
14
14
15
15
15
15
15
15
15
15
15
15
15
16
16
16
n
14
15
16
17
18
19
20
21
22
23
24
25
15
16
17
18
19
20
21
22
23
24
25
16
17
18
5% 21/2% 1% 1/2%
10% 5% 2% 1%
1 – tail
2 – tail
m
16
16
16
16
16
16
16
17
17
17
17
17
17
17
17
17
18
18
18
18
18
18
18
18
n
19
20
21
22
23
24
25
17
18
19
20
21
22
23
24
25
18
19
20
21
22
23
24
25
5% 21/2% 1% 1/2%
10% 5% 2% 1%
71
120 115 109 105
166 160 152 147
237 228 218 210
73
125 119 113 109
171 164 156 151
243 234 223 215
76
129 123 116 112
176 169 161 155
249 239 228 220
79
133 127 120 115
182 174 165 159
255 245 233 225
81
138 131 124 119
187 179 170 163
261 251 238 230
84
142 135 127 122
192 183 174 168
267 256 244 235
86
146 139 131 125
197 188 178 172
273 262 249 240
89
150 143 134 129
202 193 183 176
249 240 230 223
92
155 147 138 132
207 198 187 180
255 246 235 228
94
159 151 142 136
212 203 192 184
262 252 241 234
97
163 155 145 139
218 207 196 188
268 258 246 239
99
168 159 149 142
223 212 200 192
274 264 252 244
102
172 163 153 146
192 184 176 171
281 270 258 249
105
176 167 156 149
197 190 181 175
287 276 263 255
107
142 136 130 125
203 195 186 180
294 282 269 260
110
147 141 134 129
208 200 190 184
300 288 275 265
87
152 145 138 133
214 205 195 189
280 270 259 252
90
156 150 142 136
220 210 200 193
287 277 265 258
93
161 154 146 140
225 216 205 198
294 283 271 263
96
166 158 150 144
231 221 210 202
301 290 277 269
99
171 163 154 148
236 226 214 207
307 296 283 275
102
175 167 158 151
242 231 219 211
314 303 289 280
105
180 171 162 155
248 237 224 216
321 309 295 286
108
185 176 166 159
219 211 202 196
328 316 301 292
111
189 180 170 163
225 217 207 201
114
194 185 174 166
231 222 212 206
117
199 189 178 170
120
123
126 For larger values of m, n it is usually adequate to use a Normal approximation, with continuity correction,
129
1
1
1
with mean –2 mn + –2 m(m + 1) and variance ––
mn(m + n + 1)
12
1 – tail
2 – tail
5% 21/2% 1% 1/2%
10% 5% 2% 1%
m
19
19
19
19
19
19
19
20
20
20
20
20
20
21
21
21
21
21
22
22
22
22
23
23
23
24
24
25
313
320
328
335
342
350
357
348
356
364
371
379
387
385
393
401
410
418
424
432
441
450
465
474
483
507
517
552
n
19
20
21
22
23
24
25
20
21
22
23
24
25
21
22
23
24
25
22
23
24
25
23
24
25
24
25
25
303
309
316
323
330
337
344
337
344
351
359
366
373
373
381
388
396
404
411
419
427
435
451
459
468
492
501
536
291
297
303
310
316
323
329
324
331
337
344
351
358
359
366
373
381
388
396
403
411
419
434
443
451
475
484
517
283
289
295
301
307
313
319
315
322
328
335
341
348
349
356
363
370
377
386
393
400
408
424
431
439
464
472
505
CRITICAL VALUES FOR THE WILCOXON RANK SUM 2-SAMPLE TEST
29
m
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
11
11
11
11
11
11
11
11
11
11
11
11
11
11
11
5% 21/2% 1% 1/2%
10% 5% 2% 1%
Critical values for the Wilcoxon Single Sample and Paired Sample tests
5%
10%
21/2%
5%
1%
2%
%
1%
1/
2
n
1 – tail
2 – tail
5%
10%
21/2%
5%
1%
2%
%
1%
1/
2
n
Group Size
Action Lines
Warning Lines
n
D1
D2
D3
D4
2
0.00
4.12
0.04
2.81
75
83
91
100
109
2.18
–
–
–
–
84
92
101
110
120
0.18
–
–
–
–
98
107
116
126
137
2.99
–
–
–
–
110
119
130
140
151
0.04
–
–
–
0
26
27
28
29
30
3
2
3
4
5
4
0.10
2.58
0.29
1.94
5
0.16
2.36
0.37
1.80
6
0.21
2.22
0.42
1.72
6
7
8
9
10
2
3
5
8
10
0
2
3
5
8
–
0
1
3
5
–
–
0
1
3
31
32
33
34
35
163
175
187
200
213
147
159
170
182
195
130
140
151
162
173
118
128
138
148
159
7
0.26
2.12
0.46
1.66
8
0.29
2.05
0.50
1.62
9
0.33
1.99
0.52
1.58
10
0.35
1.94
0.54
1.55
30
11
12
13
14
15
13
17
21
25
30
10
13
17
21
25
7
9
12
15
19
5
7
9
12
15
36
37
38
39
40
227
241
256
271
286
208
221
235
249
264
185
198
211
224
238
171
182
194
207
220
16
17
18
19
20
35
41
47
53
60
29
34
40
46
52
23
27
32
37
43
19
23
27
32
37
41
42
43
44
45
302
319
336
353
371
279
294
310
327
343
252
266
281
296
312
233
247
261
276
291
21
22
23
24
25
67
75
83
91
100
58
65
73
81
89
49
55
62
69
76
42
48
54
61
68
46
47
48
49
50
389
407
426
446
466
361
378
396
415
434
328
345
362
379
397
307
322
339
355
373
n(n + 1)
For larger values of n, the Normal approximation with mean ––––––– ,
4
n(n + 1)(2n + 1)
variance ––––––––––––– should be used for T = min [P,Q].
24
The action and warning lines are obtained by multiplying the values in the
table by the mean range of the values obtained from the process.
CRITICAL VALUES FOR THE WILCOXON SINGLE SAMPLE AND PAIRED SAMPLE TESTS
SHEWHART CHART: ACTION AND WARNING LINES
1 – tail
2 – tail
Action and Warning lines for Shewhart Chart for Ranges
Random Numbers
35335
36545
80195
68554
22164
22379
08533
68554
73208
45059
10532
98804
00633
26598
26739
72151
81751
93118
99822
60413
19059
04950
94608
48526
13814
24386
56693
20189
64049
40214
71329
59305
30914
90952
05161
92325
83855
86787
93548
83980
27993
60544
62681
54466
30871
65746
00149
84105
92314
90467
69055
14986
87317
14939
69092
10989
45902
77226
19919
60679
96803
59948
16664
64984
04412
79920
37723
90447
77405
40176
11516
12133
84319
94768
29795
16513
99126
18307
45035
42457
94355
08205
37725
32848
78342
54346
33406
89923
45518
68712
24413
17232
50818
92295
59002
99070
77339
54796
58355
40737
61327
01422
03374
19144
13472
62796
23117
49807
43490
50490
84262
53582
66450
77677
37774
41531
53867
67301
43243
71636
Estimation of standard deviation from range
n
an
n
an
n
an
n
an
2
0.8862
5
0.4299
8
0.3512
11
0.3152
3
0.5908
6
0.3946
9
0.3367
12
0.3069
4
0.4857
7
0.3698
10
0.3249
13
0.2998
3
2
4
1
2
4
2
4
2
3
1
1
2
4
2
2
2
3
4
2
1
2
1
2
4
1
3
2
1
1
1
2
3
4
3
3
1
4
1
4
1
3
2
3
4
3
1
3
3
2
4
4
3
1
1
3
3
1
3
3
4
4
2
3
3
4
4
4
4
3
4
4
4
2
2
2
2
3
2
1
2
1
3
2
3
1
4
1
4
1
3
2
1
3
4
1
4
4
2
4
2
3
4
4
1
2
1
1
3
2
3
1
1
1
4
4
3
1
4
2
4
4
1
4
1
2
3
2
1
4
2
3
4
2
3
4
1
2
1
1
3
1
3
1
2
3
2
3
2
4
2
3
2
3
1
1
4
2
3
3
2
4
3
1
1
2
4
3
3
3
4
2
3
2
1
1
2
3
4
4
2
4
1
1
4
3
1
4
4
4
3
3
4
2
4
4
2
3
1
3
4
3
1
4
2
4
1
1
4
4
3
3
1
4
3
3
4
1
2
1
4
1
2
3
3
2
4
2
3
2
2
4
3
3
3
3
4
4
4
1
3
1
2
2
4
3
2
2
2
2
2
1
4
1
4
4
3
2
1
2
3
2
3
4
2
1
3
3
2
1
1
1
1
4
2
1
1
1
3
2
1
2
4
3
3
1
3
4
1
1
1
4
2
3
2
2
1
4
3
3
1
3
4
2
1
4
2
1
1
3
4
2
2
1
1
2
3
2
2
4
4
3
1
4
1
3
3
3
1
3
2
1
4
4
3
2
3
4
4
2
4
1
4
2
1
2
1
3
2
4
4
4
3
4
2
3
1
4
2
4
3
1
4
3
3
4
4
2
4
4
1
4
2
2
2
3
2
1
2
3
1
3
3
4
4
1
4
4
3
2
1
3
4
3
3
2
3
1
1
3
2
4
3
2
2
1
1
1
2
1
4
2
3
3
4
1
1
2
3
4
3
4
2
3
3
4
2
2
1
1
2
1
2
1
4
1
4
3
4
1
1
2
2
1
4
2
2
4
3
2
3
3
1
1
1
4
4
3
1
1
2
2
1
1
2
3
3
1
4
3
3
2
1
2
1
4
2
2
3
2
RANDOM NUMBERS AND RANDOM PERMUTATIONS
ESTIMATION OF STANDARD DEVIATION FROM RANGE
31
68236
62385
64058
64822
17716
03928
11021
01830
36782
58158
91239
27073
81501
64374
29896
38996
73936
18795
76816
12091
41538
12909
49185
37771
22532
60132
23784
03081
51273
03281
Random permutations (size 4)
Random permutations (size 5)
2
5
5
5
2
5
3
2
4
5
4
1
4
1
2
2
2
4
2
2
1
3
4
1
4
3
1
2
5
2
5
2
1
1
3
3
1
5
4
5
3
1
3
3
3
1
4
5
1
1
1
5
1
3
4
4
4
1
1
1
4
2
3
5
5
4
2
4
2
4
4
1
3
5
4
5
4
1
1
1
4
3
2
4
1
4
1
4
5
3
5
3
3
2
1
3
5
5
5
5
3
4
5
3
1
1
4
1
1
1
1
4
4
3
5
4
2
2
5
3
1
4
1
1
4
2
5
3
3
4
2
4
5
4
5
1
1
2
3
3
5
1
1
2
3
2
3
3
3
3
2
3
2
4
1
2
3
4
2
4
4
3
2
1
5
5
4
2
1
3
5
3
3
4
1
1
4
1
1
2
1
1
3
1
3
1
4
5
2
2
5
1
1
3
3
4
5
1
4
1
2
1
1
5
3
4
5
4
2
5
2
1
1
2
5
5
5
5
5
3
4
3
5
3
5
2
3
2
5
5
4
5
2
5
4
1
3
5
3
2
3
2
4
3
4
3
2
5
3
2
3
4
5
3
3
2
3
3
3
5
3
5
2
5
4
3
1
3
3
3
2
3
5
4
2
5
1
2
5
3
5
4
3
4
1
2
3
3
5
1
1
2
2
5
4
4
1
4
4
1
5
4
1
2
1
4
2
1
4
1
3
2
3
2
5
3
4
3
1
5
1
5
5
2
2
1
1
1
4
4
4
5
4
1
2
3
2
2
2
4
2
2
4
4
2
5
5
4
1
4
1
4
4
1
1
2
2
4
2
4
3
5
2
1
2
5
2
3
4
5
3
1
1
4
1
3
5
3
1
5
5
5
2
4
4
5
2
4
2
3
3
3
2
2
3
5
5
1
1
2
1
3
1
4
5
1
5
4
1
4
2
4
2
5
3
4
3
5
2
1
2
1
5
1
1
4
1
3
4
1
5
4
5
1
5
3
1
5
5
3
5
2
5
3
4
4
3
1
2
2
1
5
3
1
5
1
1
4
5
4
4
4
4
5
5
3
3
5
3
5
4
5
4
5
2
1
3
3
2
5
4
4
4
2
3
3
4
2
5
1
5
2
5
3
2
5
4
1
4
3
3
2
1
3
3
1
5
2
5
4
1
2
1
4
1
2
4
2
4
1
2
1
3
5
1
2
1
5
3
3
4
3
4
2
4
2
2
2
3
2
1
3
3
2
2
2
4
1
1
2
2
1
3
3
4
4
2
4
3
4
5
7
10
5
5
3
10
8
5
10
6
10
6
9
9
3
7
1
7
1
10
9
4
3
9
1
3
4
9
9
5
2
9
5
5
6
4
1
6
5
8
2
1
2
1
2
6
4
9
8
9
8
3
4
3
2
2
10
9
5
4
10
8
8
3
4
7
9
5
3
1
8
10
8
10
4
7
7
9
9
1
4
2
6
8
7
5
7
2
7
5
2
4
7
7
9
10
3
2
6
7
6
2
5
4
7
8
10
10
6
3
5
8
10
7
3
8
8
1
2
6
8
4
7
4
6
9
9
4
4
7
6
8
10
8
4
9
4
1
9
5
5
9
2
7
3
9
2
4
4
6
4
6
3
2
2
1
9
3
7
7
6
9
1
9
10
7
10
3
5
2
3
5
6
2
1
1
6
6
2
8
7
10
9
1
8
5
3
1
7
7
6
7
6
4
10
6
3
7
10
9
1
3
3
6
8
4
6
7
9
10
7
10
2
5
6
4
2
3
8
3
3
1
7
6
10
4
1
6
2
8
10
2
9
8
5
10
4
2
3
2
3
7
10
3
5
3
3
1
3
8
4
2
1
4
10
3
9
10
3
1
1
7
4
10
5
2
8
7
5
9
1
3
1
3
7
2
2
5
4
10
10
5
9
10
1
1
1
6
6
3
9
9
5
6
7
5
8
4
7
6
8
5
6
2
6
6
5
8
1
10
3
1
7
1
8
9
6
8
6
3
5
6
4
7
4
2
2
8
2
4
5
7
8
1
8
8
7
8
4
2
4
3
10
5
9
10
6
3
10
2
7
4
4
9
9
5
5
4
1
4
9
8
8
2
9
8
5
10
1
1
1
1
3
10
5
6
5
5
10
9
2
6
1
8
2
1
7
2
8
4
9
5
2
6
1
3
10
10
8
8
3
9
6
8
6
9
9
4
5
10
7
9
2
8
8
9
2
10
10
4
2
9
10
3
1
3
7
2
2
3
2
8
7
9
9
8
2
7
8
2
5
1
3
4
9
6
1
6
6
6
3
6
6
4
1
10
9
2
6
8
3
2
5
4
3
4
5
5
4
5
4
5
3
7
8
10
10
9
1
5
8
3
10
10
2
1
6
10
2
8
6
1
5
7
9
8
8
1
3
7
1
8
8
1
4
5
10
7
9
8
6
6
10
8
1
7
4
3
9
7
6
10
9
3
5
4
8
8
7
2
5
3
1
9
10
1
4
7
1
9
8
1
2
2
2
8
2
3
3
1
8
4
5
3
10
4
1
6
2
9
1
7
7
9
8
2
2
1
1
9
2
4
9
1
3
4
5
4
8
6
9
10
6
9
6
9
1
6
5
2
7
1
6
6
5
6
9
4
3
10
9
4
4
5
3
5
3
9
4
5
8
8
8
10
5
2
6
9
9
10
5
6
9
5
10
7
8
1
1
4
1
7
4
5
3
3
7
10
7
1
2
6
5
7
10
10
10
7
8
4
9
10
7
5
4
6
1
8
7
3
6
3
4
8
9
10
6
8
7
7
9
10
2
1
2
5
6
1
4
4
10
5
1
6
4
7
5
5
9
1
10
2
3
6
2
7
10
5
2
5
10
5
1
10
5
1
9
10
8
6
3
2
9
4
7
2
3
2
10
6
7
8
2
1
7
3
7
3
3
7
4
5
4
2
6
3
3
3
4
2
4
7
7
7
8
2
3
4
6
9
10
3
8
10
4
9
8
5
5
3
4
2
8
2
1
8
4
2
10
3
1
7
10
3
7
5
7
6
5
1
7
4
3
6
7
6
4
9
10
10
5
9
1
2
6
1
5
8
6
RANDOM PERMUTATIONS
32
5
2
4
2
5
3
2
1
2
2
3
2
2
5
3
5
3
3
4
4
2
5
2
4
2
5
5
5
4
5
3
5
5
2
2
1
5
3
3
2
Random permutations (size 10)